Environment for creative processing of text and numerical data

Permutation test

This is an abbreviated version of a Finnish teaching program made in 1998.

HH - HT game (analysis)

This demo in YouTube

HH - HT game (simulation)

This demo in YouTube

Solving a Survo puzzle

This demo in YouTube (created in 2010)

I presented Survo Puzzle in 2006. More information is available on the home page.

Lines going through 3 points in a 9x9 grid

This demo in YouTube

Solving a Survo puzzle by the swapping method

This demo in YouTube

    A  B  C  D  E
 1  *  *  *  *  * 41
 2  *  *  *  *  * 28
 3  *  *  *  *  * 51
   34  8 13 28 37

Prime numbers listed by a sucro

The prime numbers are found by using a variation of the 'trial division' method:

Ulam spiral in color

Background information about the Ulam spiral in Wikipedia, for example.

When making the spiral the main task is to map values of n to x,y coordinates.
I derived the formulas

x(n)=x(n-1)+sin(mod(int(sqrt(4*(n-2)+1)),4)*pi/2)
y(n)=y(n-1)-cos(mod(int(sqrt(4*(n-2)+1)),4)*pi/2)

by observing that the turning points of the spiral may be described in this way

           12 11 11 11 11 11 11
           12  8  7  7  7  7 10  .
           12  8  4  3  3  6 10  .
           12  8  4  1  2  6 10 14
           12  8  5  5  5  6 10 14
           12  9  9  9  9  9 10 14
           13 13 13 13 13 13 13 14

1,2,3,3,4,4,5,5,5,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,9,...
with a general term a(n)=int(sqrt(4n+1)), n=0,1,2,...

Comparing correlations in two samples by a randomized test

• See also:
  Statistical tests in Survo
  Permutation test
  Fisher's exact test for contingency tables
  Comparing two samples
  Fisher transformation.

Age pyramid (Finland 2009)

The age pyramid (TYPE=PYRAMID) is one of types of bar charts in Survo.

Letter frequencies in Shakespeare's Sonnets

Shakespeare's 154 Sonnets were imported from
http://www.shakespeares-sonnets.com.

The same textual data is studied in demos Shakespeare's Sonnets as a Markov chain and
The most common words in Shakespeare's Sonnets.

Letter  Sonnets English Difference
          %       %
a       6.8     8.2    -1.4
b       1.7     1.5     0.2
c       1.8     2.8    -1.0
d       3.8     4.3    -0.5
e      12.5    12.7    -0.2
f       2.3     2.2     0.1
g       1.9     2.0    -0.1
h       7.0     6.1     0.9
i       6.4     7.0    -0.6
j       0.1     0.2    -0.1
k       0.8     0.8     0.0
l       4.2     4.0     0.2
m       2.9     2.4     0.5
n       6.2     6.7    -0.5
o       7.8     7.5     0.3
p       1.4     1.9    -0.5
q       0.1     0.1     0.0
r       5.7     6.0    -0.3
s       6.8     6.3     0.5
t       9.9     9.1     0.8
u       3.2     2.8     0.4
v       1.3     1.0     0.3
w       2.6     2.4     0.2
x       0.1     0.2    -0.1
y       2.7     2.0     0.7
z       0.0     0.1    -0.1

Shakespeare's Sonnets as a Markov chain

Shakespeare's 154 Sonnets were imported from http://www.shakespeares-sonnets.com.
The same textual data is studied in demos Letter frequencies in Shakespeare's Sonnets and
The most common words in Shakespeare's Sonnets.

The simulations were made according to a technique presented by Claude Shannon in
A Mathematical Theory of Communication (1948).

Linear regression analysis by orthogonalization

Most common words in Shakespeare's Sonnets

Shakespeare's 154 Sonnets were imported from http://www.shakespeares-sonnets.com.
The same textual data is studied in demos Letter frequencies in Shakespeare's Sonnets and
Shakespeare's Sonnets as a Markov chain.

The most important tools were the WORDS, STAT, and SORT commands.

The order of the most common words differs from that of common English for obvious reasons.

Discriminant analysis of Iris flower data set

MATRIX DISCRXR.M
Correlations_between_variables_and_discriminators
///        Discr1   Discr2
Sepal_L  -0.79189 -0.21759
Sepal_W   0.53076 -0.75799
Petal_L  -0.98495 -0.04604
Petal_W  -0.97281 -0.22290

Cluster analysis of Iris flower data set

F1 connections

"Rotated arrowheads"

This demo in YouTube

Influence curves for the correlation coefficient

This demo in YouTube

The plotting setup

PLOT z(x,y)=abs(r*(1-w)+u*v)/w
  u=sqrt(n/(n*n-1))*(x-mx)/sx   means mx,my
  v=sqrt(n/(n*n-1))*(y-my)/sy   standard deviations sx,sy
  w=sqrt((1+u*u)*(1+v*v))
TYPE=CONTOUR SCREEN=NEG ZSCALING=20,0

I presented this example among others in my talk about Survo in Compstat 1992 (Neuchâtel).
This graph was used by the organizers of Compstat as a cover page (upside down!!)
in the proceedings of the symposium.

See also

Colored texts in bar/pie charts

"Hello World!"

Virtual keyboard

Cycloid

Prime factors of numbers m^n-1

*TUTSAVE MPN
/ /MPN m_max,n                              / SM 4.12.2010
/ assuming that n is a prime number
/ computes the prime factors of numbers (m^n-1)/(m-1) for
/ m=2,3,...,m_max
/ and represents them in the form c*n+1.
/ If m-1 divides n, the smallest factor is n.
/
/ See: ../papers/MustonenPrimes.pdf
/
/ def Wmax=W1 Wn=W2 Wm=W3 Wprod=W4 Wc=W5 Wfactor=W6 Wpow=W7
/
*{init}{tempo 0}{disp off}{Wm=2}{R}
*int(exp(log(9000000000000000)/{print Wn}))={act}{l} {save word Wc}
*{line start}{erase}{u}{disp on}{tempo 2}
- if Wmax <= Wc then goto A
*{Wmax=Wc}
+ A: {R}
*({print Wm}^{print Wn}-1)/({print Wm}-1)={act}{l} {line end}
*(10:factors)={act}
/ Remove text "(10:factors)=":
*{l13}{del12}{r}{ref set 1}
/
*{save line Wprod}{erase}{R}{print Wprod}{line start}
/ Replace *'s by spaces:
+ B: {r}{save char Wc}
- if Wc '=' {sp} then goto C
- if Wc '<>' * then goto B
/ Replace * by a space:
* {goto B}
+ C:
/ Each factor to a separate line:
*{home}{u}{ins line}TRIM 1{act}{del line}{home}
*{save char Wc}
- if Wc '<>' {sp} then goto D
*{del line}
/
+ D: {save word Wfactor}{Wpow=1}
- if Wfactor = 0 then goto G
- if Wfactor > Wn then goto D1
/ Wfactor is n
*{form}{goto F}
+ D1:
/ Search for ^
+ D2: {r}{save char Wc}
- if Wc '=' ^ then goto D3
- if Wc '<>' {sp} then goto D2 else goto D4
+ D3:  {save word Wpow}{home}{save word Wfactor}
+ D4: {line start}{erase}({print Wfactor}-1)/{print Wn}={act}
/
*{l} {save word Wc}{line start}{erase}({print Wc}*{print Wn}+1)
/
*{line start}{save word Wfactor}
+ F: {ref jump 1}{write Wfactor}
- if Wpow = 1 then goto F2
*^{print Wpow}
+ F2: *{ref set 1}{R}{del line}{goto D}
+ G: {ref jump 1}{l}{del}
/
- if Wm = Wmax then goto E
*{Wm=Wm+1}{goto A}
+ E: {end}

Some properties of Magic Squares

This demo is created by Kimmo Vehkalahti.

Functions of the Survo matrix interpreter are shown in connection with Magic Squares.

Some further properties of Magic Squares

This demo is created by Kimmo Vehkalahti.

Functions of the Survo matrix interpreter are shown in connection with Magic Squares.

Solving linear equations

A system of linear equations

X1+X2=27
X2+X3=32
X3+X4=32
X1+X3=25

MATRIX A
///  X1 X2 X3 X4
r12   1  1  0  0
r23   0  1  1  0
r34   0  0  1  1
r13   1  0  1  0

MATRIX B
/// freq
r12   27
r23   32
r34   32
r13   25

Polynomial regression

Marking columns by SHADOW SET

Hunting quanta

                         n
    phi(q) = sqrt(2/n)* SUM cos(2*pi*eps(i)/q)                (Kendall 1974)
                        i=1

                      n
   ss(q_1,...,q_k) = SUM min[g(x_i,q_1)^2,...,g(x_i,q_k)^2]       (SLS 2005)
                     i=1

Survopoint display mode

e 30 159 S
*    The road to hell is paved with good intentions.
*    He laughs best who laughs last.
*    A smooth sea never made a skilled mariner.
*    Truth is stranger than fiction.
*    A friend to all is a friend to none.
*    Be swift to hear, slow to speak.
*    Knowledge in youth is wisdom in age.
 ...

Four-dimensional cube

 65 *
 66 *The following sucro command activates all commands having a '+' in the
 67 *control column and thus the final graph will be automatically created:
 68 *
 69 */ACTIVATE +  (Activated commands are displayed here in red.)
 70 *It is possible to draw each 2-dimensional projection of a 4-dimensional
 71 *cube as a single line graph of edges since the degree of each vertex
 72 *is 4. Then there exists an Eulerian circuit where each edge is
 73 *traversed just once.
 74 *Consider the cube in a 4-dimensional space so that vertices have
 75 *coordinates (x_1,x_2,x_3,x_4) where each x_i is either 0 or 1.
 76 *Then the following matrix gives an Eulerian circuit in this
 77 *4-dimensional cube:
 78 *
 79 *MATRIX C4 ///
 80 *0 0 0 0
 81 *0 0 0 1
 82 *0 0 1 1
 83 *0 0 1 0
 84 *0 0 0 0
 85 *0 1 0 0
 86 *0 1 0 1
 87 *0 1 1 1
 88 *0 1 1 0
 89 *0 1 0 0
 90 *1 1 0 0
 91 *1 1 0 1
 92 *0 1 0 1
 93 *0 0 0 1
 94 *1 0 0 1
 95 *1 1 0 1
 96 *1 1 1 1
 97 *0 1 1 1
 98 *0 0 1 1
 99 *1 0 1 1
100 *1 0 0 1
101 *1 0 0 0
102 *1 1 0 0
103 *1 1 1 0
104 *0 1 1 0
105 *0 0 1 0
106 *1 0 1 0
107 *1 0 1 1
108 *1 1 1 1
109 *1 1 1 0
110 *1 0 1 0
111 *1 0 0 0
112 *0 0 0 0
113 *
114 +MAT SAVE C4
115 +MAT TRANSFORM C4 BY X#-0.5   / Centering (0,1) -> (-0.5,0.5)
116 +MAT CLABELS "X" TO C4        / Column labels X1,X2,X3,X4
117 *
118 *The regular 2-dimensional projections of this hypercube are plain
119 *squares and thus not very interesting.
120 *
121 *A better view is obtained by making an "arbitrary" 4-dimensional
122 *rotation:
123 *
124 +MAT T=ZER(4,4)
125 +MAT TRANSFORM T BY sin(31*I#*J#)  / "arbitrary" T
126 *
127 +MAT GRAM-SCHMIDT DECOMPOSITION OF T TO Q,R  / Orthogonalization of T
128 +MAT K=C4*Q                  / Rotation of the hypercube by orthogonal Q
129 +MAT CLABELS "dim" TO K      / Column labels dim1,dim2,dim3,dim4
130 *
131 *Combining the rotated and original cube into one matrix KB:
132 *
133 +MAT KB=ZER(33,8)
134 +MAT KB(1,1)=K
135 +MAT KB(1,5)=C4
136 *
137 *.......................................................................
138 *Plotting all six 2-dimensional projections separately:
139 *
140 *SIZE=1000,1000 SCALE=-1,1 HEADER= XDIV=0,1,0 YDIV=0,1,0 FRAME=3
141 *FRAMES=F F=0,0,1000,1000 PEN=[SwissB(30)]
142 *XLABEL= YLABEL= LINE=[line_type(2)][line_width(0.2)],1 TEXTS=T
143 *
144 +PLOT KB.MAT,dim1,dim2 / DEVICE=PS,A12.PS T=1_-_2,750,50
145 +PLOT KB.MAT,dim1,dim3 / DEVICE=PS,A13.PS T=1_-_3,750,50
146 +PLOT KB.MAT,dim1,dim4 / DEVICE=PS,A14.PS T=1_-_4,750,50
147 +PLOT KB.MAT,dim2,dim3 / DEVICE=PS,A23.PS T=2_-_3,750,50
148 +PLOT KB.MAT,dim2,dim4 / DEVICE=PS,A24.PS T=2_-_4,750,50
149 +PLOT KB.MAT,dim3,dim4 / DEVICE=PS,A34.PS T=3_-_4,750,50
150 *
151 *.......................................................................
152 *Plotting two opposite 3-dimensional cubes in different colors (blue and
153 *red):
154 *
155 *SIZE=1000,1000 SCALE=-1,1 HEADER= XDIV=0,1,0 YDIV=0,1,0 FRAME=0
156 *XLABEL= YLABEL=
157 *               *blue=[color(1,1,0,0)],1 *red=[color(0,1,1,0)],1
158 +PLOT KB.MAT,dim1,dim2 / DEVICE=PS,B12.PS IND=X1,-0.5 LINE=*blue
159 +PLOT KB.MAT,dim1,dim2 / DEVICE=PS,C12.PS IND=X1,0.5  LINE=*red
160 +PLOT KB.MAT,dim1,dim3 / DEVICE=PS,B13.PS IND=X1,-0.5 LINE=*blue
161 +PLOT KB.MAT,dim1,dim3 / DEVICE=PS,C13.PS IND=X1,0.5  LINE=*red
162 +PLOT KB.MAT,dim1,dim4 / DEVICE=PS,B14.PS IND=X1,-0.5 LINE=*blue
163 +PLOT KB.MAT,dim1,dim4 / DEVICE=PS,C14.PS IND=X1,0.5  LINE=*red
164 +PLOT KB.MAT,dim2,dim3 / DEVICE=PS,B23.PS IND=X1,-0.5 LINE=*blue
165 +PLOT KB.MAT,dim2,dim3 / DEVICE=PS,C23.PS IND=X1,0.5  LINE=*red
166 +PLOT KB.MAT,dim2,dim4 / DEVICE=PS,B24.PS IND=X1,-0.5 LINE=*blue
167 +PLOT KB.MAT,dim2,dim4 / DEVICE=PS,C24.PS IND=X1,0.5  LINE=*red
168 +PLOT KB.MAT,dim3,dim4 / DEVICE=PS,B34.PS IND=X1,-0.5 LINE=*blue
169 +PLOT KB.MAT,dim3,dim4 / DEVICE=PS,C34.PS IND=X1,0.5  LINE=*red
170 *
171 *Coloring the projections:
172 +EPS JOIN K12,A12,B12,C12
173 +EPS JOIN K13,A13,B13,C13
174 +EPS JOIN K14,A14,B14,C14
175 +EPS JOIN K23,A23,B23,C23
176 +EPS JOIN K24,A24,B24,C24
177 +EPS JOIN K34,A34,B34,C34
178 *
179 *Entering coordinates for projections in the final setup:
180 *K12=K12,0,2000
181 *K13=K13,0,1000   K23=K23,1000,1000
182 *K14=K14          K24=K24,1000,0      K34=K34,2000,0
183 *
184 *Combining the parts:
185 +EPS JOIN CUB4,K12,K13,K14,K23,K24,K34
186 *
187 *Creating the result
188 *as a PostScript file:
189 +PRINT CUR+1,X TO Cube4.PS
190 % 1500
191 - picture CUB4.PS,*,*,0.47,0.47
192 X
193 *Making and displaying
194 *a PDF file Cube4.PDF:
195 +/GS-PDF Cube4.PS
196 *
197 *The result is converted
198 *and displayed here
199 *as an EMF file.
200 *

'Word' processing by mouse

Copies of various items in the edit field can be made in various ways.

Immediately after the first copy, more copies can be made by the leftmost mouse button.
If the mouse is pointing at a blank space between existing 'words', the copy is inserted between these words.
If the mouse is pointing at a 'word' (a non-blank character), this 'word' is replaced by the copy.
3. The copying process is terminated by the DEL key.

- - - - - - - - - -

When using Survo there is no absolute need for working with a mouse. For many people, operating with a classical mouse is a nuisance causing physical stress and pain.

For many years, at last for me, a 'RollerMouse' (coupled with the splendid IBM PC/AT keyboard) has been much better as a pointing device. When using it there is no need to move hands or wrist away from the keyboard. All mouse functions can be executed by minimal moves of the fingertips.

---

Edges and diagonals of a regular n-sided polygon

Example #77 (Start the demo by clicking the picture!)

By means of Survo and Mathematica I have found certain properties related to lengths of edges and diagonals of a regular n-sided polygon inscribed in a unit circle.
In particular, in this demo it was shown experimentally that in the regular heptagon the squared lengths of the chords are roots of an algebraic equation
X^3-7*X^2+14*X-7=0 (known already by Kepler).

In general, I found experimentally that the squared lengths of the diagonals in a regular n-sided polygon are roots of an algebraic equation with coefficients as simple expressions of binomial coefficients.
These results were formally proved in my paper with Pentti Haukkanen and Jorma Merikoski.

See also
http://www.survo.fi/papers/Polygons2013.pdf
http://www.survo.fi/papers/Roots2013.pdf.
and
Mustonen, S., Haukkanen, P., Merikoski, J. (2014).
Some polynomials associated with regular polygons.
Acta Univ. Sapientiae, Mathematica, 6, 2, 178-193.


More extensive demos about the same subject:
Equation for the sum of chord lengths in a regular polygon
Regular polygons: Solving riddle of q coefficients
Regular polygons: Testing roots

---

Examples from a presentation in 1987

Example #78 (Start the demo by clicking the picture!)

This demo in YouTube

This is partial reproduction of my talk "Editorial approach in statistical computing"
in the Second International Tampere Conference in Statistics (1987).
This demo gives two examples about usage of Survo.
In the original video

many details related to these examples are difficult to see.

The last example 'Estimation of a circle' of my talk is also available in YouTube.

The final paper in conference proceedings does not include these examples.

---

Thurstone's box problem

Example #79 (Start the demo by clicking the picture!)

This demo in YouTube

Thurstone's problem is presented in a more general form. Values of the derived variables have substantial 'measurement' errors.

The Thurstone's original experiment is described, for example, in Richard L. Gorsuch: Factor Analysis pp. 10-11.

Finding recursive formula for number of grid lines

This demo in YouTube and also as a flash demo.

See also

Testing the correlation coefficient

This demo in YouTube

Matrix interpreter (regression analysis)

This demo in YouTube

This demo was a part of my talk Matrix computations in Survo
in The Eighth International Workshop on Matrices and Statistics (1999)
at the University of Tampere, Finland.

FILE SAVE MAT <matrix_file> TO <Survo_data_file>

MAT SAVE DATA <Survo_data> TO <matrix_file>

"Origin of Species" 2

This demo in YouTube

GPLOT x(t)=X0+R*(sin(t)+r*sin(A*t)+r^2*sin(B*t)),
      y(t)=Y0+R*(cos(t)+r*cos(A*t)+r^2*cos(B*t))

r=(sqrt(5)-1)/2 s=1/(1+r+r^2) R=3*r/2
t=0,2*pi,pi/1000 pi=3.141592653589793
SCALE=-4,4 OUTFILE=A
XDIV=0,1,0 YDIV=0,1,0 SIZE=381,381 HEADER= FRAME=3 MODE=381,381
i=0(1)3
a=5
b=15 T1=5;15,170,170 TEXTS=T1
PEN=[color(0.2,0.4,1,0)][SwissB(20)]
FILL(-2)=0.9,0.6,0.3,0
LINETYPE=[line_width(2)],1 SLOW=300
A=x0*a+1  X0=2*x0  x0=int(sqrt(2)*(sin(pi/2*(i+0.5)))+0.5)
B=y0*b+1  Y0=2*y0  y0=int(sqrt(2)*(cos(pi/2*(i+0.5)))+0.5)
WSIZE=381,381 WHOME=800,0 WSTYLE=0 FRAMES=F F=0,0,381,381,-2

The setup of graphs corresponds to values (here A=5, B=15) in this way:
      -A,B    A,B

      -A,-B   A,-B

MATRIX AB
///  a  b C     M     Y     c     m     y     L  W   S
  1  5 15 0.200 0.400 1.000 0.900 0.600 0.300 1 40 300
  2  0  0 1.000 1.000 1.000 0.000 0.000 0.000 1  7  50
  3  1  1 1.000 1.000 1.000 0.000 0.000 0.000 1  4  20
  4  2  2 1.000 1.000 1.000 0.000 0.000 0.000 1  3  10
  5  3  3 1.000 1.000 1.000 0.000 0.000 0.000 1  2   5
... .. .. ..... ..... ..... ..... ..... ..... . .. ...
 95  9  7 0.298 0.498 1.000 0.872 0.605 0.204 1  1   0
 96  9 15 0.260 0.397 1.000 0.858 0.655 0.175 1  1   0
 97 11 32 0.323 0.374 1.000 0.917 0.711 0.211 1  1   0
 98 16 96 0.313 0.427 1.000 0.838 0.537 0.237 1  1   0
 99 45 50 0.170 0.401 1.000 0.918 0.640 0.245 1  8 100
100 10 60 1.000 1.000 1.000 0.000 0.000 0.000 1 25 300

Simulating multivariate normal distribution

This demo in YouTube

Genesis of multivariate normal distribution 1

This demo in YouTube

Genesis of multivariate normal distribution 2

This demo in YouTube

*
*Matrix Z of independent N(0,1) variables:
*MAT Z=ZER(100000,2)              / Z is a data frame.
*MAT #TRANSFORM Z BY #RAND(20140) / Fill with uniform[0,1] values,
*MAT #TRANSFORM Z BY probit(X#)   / convert to independent N(0,1) values
*
A Common specifications:
*WHOME=271,0 WSIZE=381,381 HEADER= XDIV=0,1,0 YDIV=0,1,0 XLABEL= YLABEL=
*WSTYLE=0 FRAME=3 LINETYPE=[line_width(3)],1 POINT=11 MODE=1024,1024
*SCALE=-5,5
B
*..................................
*Coordinate axes: (common backgroud for each graph)
*GPLOT X(t)=c*t,Y(t)=(1-c)*t / SPECS=A,B
*c=0,1,1 t=-9,9,9
*OUTFILE=Z0
*..................................
*Scatter diagram of two independent N(0,1) variables, 100'000 cases
*GPLOT Z.MAT,1,2 / SPECS=A,B CONTOUR=[RED],0.001,0.5
*PEN=[BLACK][SwissB(100)] TEXTS=T T=Z,20,900
*INFILE=Z0 OUTFILE=Z1
*

A related demo: Genesis of multivariate normal distribution 1

Monthly temperature and rainfall in Helsinki

This demo in YouTube

Early sound experiment on Elliott 803 computer in 1962

This demo in YouTube

Tracing a sucro program

This demo in YouTube

The program code of /SORT with comments

*TUTSAVE SORT
/ /SORT sorts numbers below the command line in ascending order.
/ Defining variables:
/ def Wfirst Wnext Wr Wc Wc2
/
/ Initialization and setting maximum speed:
*{init}{tempo 0}
*
/ Making room for the sorted list:
*{R}{ins line}
/
/ Setting a 'wall' |:
* |
/
/ Setting a space in front of the original list:
*{line start}{u}{ins} {ins}{l}{Wc=1}
/
/ Finding next number and recording its location:
+ A: {next word}{ref set 1}{save cursor Wr,Wc2}
/
/ If no next number found, going to End:
- if  Wc2 = Wc then goto End
/
/ Saving new number:
*{Wc=Wc2}{save word Wfirst}{R}
/
/ Finding next word on result line:
+ B: {next word}{save word Wnext}
/
/ If it is the wall, inserting value of Wnext to the end:
- if Wnext '=' | then goto C
/
/ Repeating from B if proper location for Wfirst not yet found:
- if Wfirst > Wnext then goto B
/
/ Writing newest number to its place in the sorted list:
+ C: {ins}{write Wfirst} {ins}{ref jump 1}{goto A}
/
/ Finishing by removing the wall and the original list:
+ End: {R}{del}{line end}{l}{del}{line start}{u}{del line}
*{end}

Finding primes by the Sieve of Erastothenes

This demo in YouTube

SET A+{write Wstart},A+{write Wmax},A,{write Wprime}

SET A+4,A+1000000,A-1,2
SET A+9,A+1000000,A-1,3
SET A+25,A+1000000,A-1,5
SET A+49,A+1000000,A-1,7

    1 *SAVE SIEVE
    2 *
    3 a/* _sieve.c 8.4.2015/SM (8.4.2015) */
    4 *
    5 *#include <stdio.h>
    6 *#include <stdlib.h>
    7 *#include <malloc.h>
    8 *#include <math.h>
    9 *#include <survo.h>
   10 *#include <survoext.h>
   11 *
   12 *unsigned int n;
   13 *char *prime;
   14 *int j,h;
   15 *
   16 *void main(argc,argv)
   17 *int argc; char *argv[];
   18 *    {
   19 *    unsigned int i,n,max,p,count,output;
   20 *    if (argc==1) return;
   21 *    s_init(argv[1]); // initializing Survo environment for this program
   22 *    if (g<2)
   23 *        {
   24 *        sur_print("\nUsage: SIEVE N [output=0,1,2]");
   25 *        WAIT; return;
   26 *        }
   27 *    n=atoi(word[1]);
   28 *    prime=(char *)malloc(n+1); // reserving space for prime indicators
   29 *    for (i=0; i<n+1; ++i) prime[i]='1'; // at the start all are primes
   30 *    max=(int)sqrt((double)n);  // max number to be tested is sqrt(n)
   31 *    p=1; output=2;
   32 *    if (g>2) output=atoi(word[2]); // selecting scope of output
   33 *    while (p<max)
   34 *       {
   35 *       ++p;
   36 *       while (prime[p]=='0') ++p; // finding next prime p
   37 *       for (i=p*p; i<=n; i+=p) prime[i]='0'; // multiples of p composite
   38 *       } // all primes found
   39 *    count=0;
   40 *    j=r1+r; new_line();
   41 *    for (i=2; i<n+1; ++i)
   42 *       if (prime[i]=='1')
   43 *          {
   44 *          ++count; // counting number of primes
   45 *          if (output==2)
   46 *              {
   47 *              h+=sprintf(sbuf+h,"%u ",i); // collecting primes on a line
   48 *              if (h>c-7) // visible line length - 7
   49 *                  {
   50 *                  out();
   51 *                  new_line();
   52 *                  }
   53 *              }
   54 *          }
   55 *    if (output>0)
   56 *          {
   57 *          if (output==2) out();
   58 *          new_line();
   59 *          sprintf(sbuf,"Number of primes < %u is %u.",n,count);
   60 *          out();
   61 *          s_end(argv[1]); // output to be catched by the editor
   62 *          }
   63 *    return;
   64 *    }
   65 *
   66 *new_line()
   67 *    {
   68 *    ++j; h=0; *sbuf=EOS; // output=2
   69 *    return(1);
   70 *    }
   71 *
   72 *out()
   73 *    {
   74 *    edwrite(space,j,1);
   75 *    edwrite(sbuf,j,1);
   76 *    return(1);
   77 *    }
   78 A
   79 *
   80 *TIME COUNT START / Continuous activation by F2 ESC
   81 *SIEVE 1000000
   82 *TIME COUNT END   0.245
   83 *2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89
   84 *97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179
   85 *181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271
   86 *277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379
   87 *383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479
   88 *487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599
   89 *601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701
   90 *709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823
   91 *827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941
   92 *947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039
   - - - - - - - - - - -
 7817 *999529 999541 999553 999563 999599 999611 999613 999623 999631 999653
 7818 *999667 999671 999683 999721 999727 999749 999763 999769 999773 999809
 7819 *999853 999863 999883 999907 999917 999931 999953 999959 999961 999979
 7820 *999983
 7821 *Number of primes < 1000000 is 78498.
 7822 *

Combining Survo operations by sucros

This demo in YouTube
This as flash demo

For example, the entire sucro code of this demo is:

 11 *
 12 */M
 13 *TUTSAVE M
 14 *{tempo -1}{init}{jump 1,1,1,1}SCRATCH {act}{line start}
 15 /
 16 *COLX W20{act}{line start}{erase}{tempo 2}{wait 100}{tempo -1}
 17 *{R}
 18 *   {form7} Combining Survo operations by sucros {R}
 19 *{R}
 20 *Sucros are at their best when the task at hand is a sequence of{R}
 21 *standard Survo operations and the same job has to be repeated{R}
 22 *many times from various initial conditions.{R}
 23 *For example, sucros have been created for performing some multistage{R}
 24 *forms of statistical analysis.{R}
 25 *Here a sucro /FACTOR is presented. It carries out the standard steps{R}
 26 *of factor analysis:{R}
 27 *{R}
 28 *1. Computing correlations CORR.M{R}
 29 *2. Computing eigenvalues by spectral decomposition of CORR.M{R}
 30 *3. The number of factors f is determined as follows:{R}
 31 *   Eigenvalues e(1)>=e(2)>=e(3)>=...{R}
 32 *   Ratios      s(i)=e(i+1)/e(i), if e(i)>=0.9, s(i)=1 else{R}
 33 *   Let e(j)>=1 and e(j+1)=<1 and s(k)=min(s(j),s(j+1),s(j+2)){R}
 34 *   Then f=k.{R}
 35 *4. Computing the maximum likelihood solution FACT.M by FACTA{R}
 36 *5. Computing the rotated factor matrix AFACT.M by ROTATE{R}
 37 *{R}
 38 */FACTOR is now applied to the dataset DECA on the 48 best athletes{R}
 39 *of the world in 1973.
 40 /
 41 *{tempo 2}{90}{R}
 42 *{d5}{tempo 0}{d14}{u19}{tempo 2}{10}
 43 *
 44 *The 10 event{tempo 0} variables will be considered
 45 * and thus set active in DECA:{tempo 2}{20}{R}
 46 *FILE ACTIVATE DECA{keys 2}{act}
 47 /
 48 *--AAAAAAAAAA--{exit}
 49 /
 50 *{keys 0}{10}{R}
 51 *Sucro /FACTOR{tempo 0} is activated with dataset DECA:
 52 *{tempo 2}{10}{R}
 53 */FACTOR DECA{keys 2}{act}{keys 0}{30}
 54 *
 55 *{d14}{20}{d14}{20}{del line}{d21}{u18}{10}
 56 /
 57 *The rotated{tempo 0} factor matrix is made
 58 * more informative by another sucro:{tempo 2}{20}
 59 *{R}
 60 */LOADFACT{keys 2}{act}{keys 0}{30}
 61 /
 62 *{d17}{u2}
 63 *Typically{tempo 0} for Survo, the work has documented itself and it may be {R}
 64 *repeated.{tempo 2}{20}
 65 * Now e.g. {tempo 0}a four-factor solution can be obtained 'manually'
 66 *{R}
 67 *and another rotation technique can be adopted:
 68 *{tempo 2}{30}{home}{5}{u55}{5}{r13}{10}4{r}35   {keys 2}{act}{keys 0}
 69 /
 70 *{10}{d}{l6}4{r}49{erase}{5}   / ROTATION=ORTHO_CLF (by Jennrich)
 71 *{keys 2}{act}{keys 0}{10}{R}
 72 *{d28}{20}{d22}{u13}{10}SCRATCH{10}{act}{10}{R}
 73 *{u2}{keys 2}{act}{keys 0}{end}
 74 *

 11 *
 12 *TUTLOAD <Survo>\S\FACTOR
 13 /    /FACTOR <data>                         / 10.6.1991/SM (13.5.1994)
 14 / or /FACTOR <data>,<number_of_factors>
 15 *{tempo -1}{init}{R}
 16 *SCRATCH {act}{home}
 17 - if W1 '=' ? then goto A
 18 - if W1 '<>' (empty) then goto S
 19 + A: /FACTOR <data>{R}
 20 *makes a factor analysis from active variables and observations of{R}
 21 *a Survo data <data>.{R}
 22 *The steps of analysis are:{R}
 23 *1. Computing correlations CORR.M{R}
 24 *2. Computing eigenvalues by spectral decomposition of CORR.M{R}
 25 *3. The number of factors f is determined as follows:{R}
 26 *   Eigenvalues e(1)>=e(2)>=e(3)>=...{R}
 27 *   Ratios      s(i)=e(i+1)/e(i), if e(i)>=0.9{R}
 28 *               s(i)=1            else.{R}
 29 *   Let e(j)>=1 and e(j+1)=<1 and s(k)=min(s(j),s(j+1),s(j+2)){R}
 30 *   Then f=k.{R}
 31 *4. Computing the maximum likelihood solution FACT.M by FACTA{R}
 32 *5. Computing the rotated factor matrix AFACT.M by ROTATE{R}
 33 *{R}
 34 *The user can also enter the number of factors f by activating{R}
 35 */FACTOR <data>,f{R}
 36 *{goto E}
 37 /
 38 + S: CORR {print W1}{act} / Correlation matrix saved as CORR.M{R}
 39 - if W2 > 0 then goto FAC
 40 *MAT SPECTRAL DECOMPOSITION OF CORR.M TO &S,&D{act}{R}
 41 *MAT DIM &S{act}{find =} {save word W3}
 42 - if W3 = 1 then goto F
 43 *{home}{erase}{ref}MAT LOAD &D,CUR+1{act}{R}
 44 *{d2}{W1=0}
 45 + Next_line: {R}
 46 *{W1=W1+1}{next word}{next word}{save word W2}
 47 - if W2 >= 1 then goto Next_line
 48 *{W4=W1-1}
 49 - if W4 < W3 then goto D
 50 + F: {R}
 51 *{ins line}Not a proper correlation matrix for factor analysis!
 52 *{goto E}
 53 + D: {}
 54 /
 55 / def We1=W3 We2=W4 We3=W5 We4=W6
 56 / def Wsmin=W7 Wf=W8 Ws=W9
 57 *{u}{save word We1}{d}{save word We2}{d}{save word We3}{d}
 58 *{save word We4}{Wsmin=We2/We1}{Wf=W1-1}
 59 - if We2 < 0.9 then goto C
 60 *{Ws=We3/We2}
 61 - if Ws > Wsmin then goto B
 62 *{Wsmin=Ws}{Wf=W1}
 63 + B: {}
 64 - if We3 < 0.9 then goto C
 65 *{Ws=We4/We3}
 66 - if Ws > Wsmin then goto C
 67 *{Wsmin=Ws}{Wf=W1+1}
 68 + C: {ref}{ref}{u}SCRATCH {act}{home}MAT &D=&D'{act}{home}{erase}MAT L
 69 *OAD &D,12.12,CUR{act}{home}{del line}{erase}MAT KILL &*{act}{home}
 70 *{erase}Eigenvalues of the correlation matrix CORR.M:{R}
 71 *{del9}{R}
 72 *{del9}{R}
 73 *{goto FAC2}
 74 /
 75 + FAC: {Wf=W2}
 76 + FAC2: {}
 77 - if Wf = 1 then goto F
 78 *FACTA CORR.M,{print Wf},END+2{act} / Factor matrix saved as FACT.M{R}
 79 *{ins line}{u}
 80 /
 81 *ROTATE FACT.M,{print Wf},END+2{act}
 82 /
 83 * / Rotated factor matrix saved as AFACT.M{R}
 84 + E: {tempo +1}{end}
 85 *

Tracing a sucro program (Finding prime numbers)

This demo in YouTube

In principle the same technique for finding prime numbers is used as in an earlier demo.
Now instead of having a long column, the integers a listed compactly on edit lines and then permitting a better chance for viewing the process.

The sucro code with additional features (in red) used for tracing in the latter part of this demo can be studied here:



In this two-window mode, sucro SIEVE is saved by a sucro command /TUTSAVE (on line 24 above) and with parameter ECHO2.

It is possible to omit echoing for selected parts of the code by control codes {-} and {+} appearing here on lines 34 and 41.
Thus the code on lines from 35 to 40 (used for finding next number after the newest prime in the list) is not echoed.

Multiple discriminant analysis in linguistic problems

This demo in YouTube

In the latter picture the vertical axis had to be reversed for a proper comparison.

My early (1965) experiment has been described recently (2013) by Steve Pepper.

Probability of Matching Column Drums (1/2)

This demo in YouTube

Probability of Matching Column Drums (2/2)

This demo in YouTube

Distance distributions in networks 1

This demo in YouTube

Distance distributions (Twin cities bridge problem)
Distance distributions (n-dimensional cube)

Distance distributions in networks 2

This demo in YouTube

In the A matrix

MAT SAVE AS A
 1  2 10 1
 2  3  5 1
 3  4  5 1
 1  5  4 1
 2  6  4 1
 3  7  4 1
 4  8  4 1
 5  6 10 1
 6  7  5 1
 7  8  5 1
 7 12  6 0
 9 10  7 1
 -  -  - -

Distance distributions in networks 3

This demo in YouTube

The exact expected value for the distance (along the edges) between
two random points on the edges of the 4-dimensional cube can be computed
as follows:
Because this 'network' is symmetric for each edge, it is sufficient
to assume that the first random point is selected from a given
edge. Each element (vertex, edge, square, cube) of this hypercube
can be denoted by a string of the form abcd where each a,b,c, and d
can be 0, 1, or x, where x covers the range (0,1). Thus for example,
0000 is the first vertex (origo), x000 is the edge from origo to
(1,0,0,0), and xxx0 and x1xx are two of the eight cubes located in the
hypercube.

The mean distances between x000 and the other edges can be
easily computed and they are presented in the following table.

edge mean     edge mean   edge mean   edge mean
x000  1/3     0x00  1     00x0  1     000x  1
x100  5/3     1x00  1     10x0  1     100x  1
x010  5/3     0x10  2     01x0  2     010x  2
x110  8/3     1x10  2     11x0  2     110x  2
x001  5/3     0x01  2     00x1  2     001x  2
x101  8/3     1x01  2     10x1  2     101x  2
x011  8/3     0x11  3     01x1  3     011x  3
x111 11/3     1x11  3     11x1  3     111x  3

The means in the first column are related to 'opposite' edges of
x000. According to terminology used in my dissertation (1964)
they are mirror point sets for x000.
It is clear that mean inside x000 is 1/3, but why the mean
distance between random points on opposite edges is of a unit square
square is 5/3 ?
     .----.
     |    |
     |    |
     .----.
Let it be x. If two random points are selected from the edges of a
unit square, the probablity that they are selected from the same
edge is 1/4, from opposite edges similarly 1/4, and from
neighbouring edges 1/2. The means are 1/3, x, and 1 respectively.
Since the total mean in unit square is 1, we have an equation
   1 = 1/4*1/3 + 1/4*x + 1/2*1
giving x=5/3. The remaining means in the first column are thereafter
easy to comprehend.

The means in the remaining three columns are obvious, since they
are either adjacent to x000 or adjacent through a route of a constant
integer length.

Now the total expected value of the distance in the entire
4-dimensional cube is calculated as

((1*2+3*5+3*8+1*11-1)/3+2*3*(1*1+2*2+1*3))/(4*2^3)=2.03125

The general structure becomes still clearer in the 5-dimensional case
where the expected value has the form

((1*2+4*5+6*8+4*11+1*14-1)/3+2*4*(1*1+3*2+3*3+1*4))/(5*2^4)=2.5291666666667
and 2.5291666666667(12:ratio)=607/240 (0.00000000000003)

On the basis of these expressions it is obvious that
in the n-dimensional case the expression for the mean distance
can be presented in the form

E(n)=((P(n+1)-1)/3+2*(n-1)*Q(n-2))/(n*2^(n-1))

Both P(n) and Q(n) are 'weighted' sums of binomial coefficients.
In fact P-sequence is an inverse binomial transform of an arithmetic
sequence 2,5,8,11,14,... and Q-sequence a similar transform of natural
numbers 1,2,3,4,5,...

The P(n) values for n=2,...,7 are

 n  P(n)
 2  1*2=2
 3  1*2+1*5=7
 4  1*2+2*5+1*8=20
 5  1*2+3*5+3*8+1*11=52
 6  1*2+4*5+6*8+4*11+1*14=128
 7  1*2+5*5+10*8+10*11+5*14+1*17=304

By consulting OEIS (The On-Line Encyclopedia of Integer Sequences)
it is found that the sequence 2,7,20,52,128,304,... is
A066373
and P(n)=(3*n-2)*2^(n-3),  n=2,3,...

The Q(n) values for n=0,1,...,5 are

 n  Q(n)
 0  1*1=1
 1  1*1+2*1=3
 2  1*1+2*2+1*3=8
 3  1*1+3*2+3*3+1*4=20
 4  1*1+4*2+6*3+4*4+1*5=48
 5  1*1+5*2+10*3+10*4+5*5+1*6=112

and OEIS tells that 1,3,8,20,48,112,... is
A001792
and Q(n):=(n+2)*2^(n-1),  n=0,1,...

By substituting expressions of P(n+1) and Q(n-2) into the
the previous formula of E(n) we obtain
E(n)=(((3*n+1)*2^(n-2)-1)/3+2*(n-1)*n*2^(n-3))/(n*2^(n-1))
and this can be simplified into the form
E(n)=n/2+(1-2^(2-n))/(6*n).

Battle over Degrees of Freedom

This demo in YouTube

Problem of minor chord in music

This demo in YouTube

HEADER=[Swiss(20)],Major_triad_(pure)
HOME=0,175 SIZE=649,174
XSCALE=0:_,10*pi:_ YSCALE=[SMALL],-3:_,3:_ pi=3.14159265
X=0,10*pi,pi/60  XDIV=29,600,20  FRAME=0
GPLOT Y(X)=sin(20*X)+sin(25*X)+sin(30*X)
               20:25:30=4:5:6
........................................................
HEADER=[Swiss(20)],Minor_triad_(pure)
HOME=0,0 SIZE=649,174
XSCALE=0:_,10*pi:_ YSCALE=[SMALL],-3:_,3:_ pi=3.14159265
X=0,10*pi,pi/60  XDIV=29,600,20  FRAME=0
GPLOT Y(X)=sin(20*X)+sin(24*X)+sin(30*X)
               20:24:30=10:12:15
........................................................

In the corresponding tempered triads the proportions 20:25:30 and
20:24:30 were replaced by 20:20*2^(4/12):20*2^(7/12) and
20:20*2^(3/12):20*2^(7/12).

At first a Survo data file TEST of 100000 observations for an integer
variable X was created by
FILE MAKE TEST,1,100000,X,2

A fundamental tone of frequency F Hz on sampling rate RATE is described
by a sinus function of the form
S(x):=sin(x*ORDER*2*pi*F/RATE)
where ORDER in the index of an observation in the data file TEST.

When F=440, RATE=44100, and pi=3.141592653589793
S(1) will represent a sample of tone A=440 Hz
and a sample of a pure A Major triad is saved in the data file TEST
as variable X by the VAR operation
VAR X=1000*(S(1)+S(5/4)+S(3/2)) TO TEST
where 1000 gives the sound volume.

Now this TEST file is converted to a WAV file MAJOR.WAV by the Survo
operation
PLAY DATA TEST,X / WAV=MAJOR
and thereafter it can be played as a sound of length
100000/RATE=2.267... seconds by activating
PLAY SOUND MAJOR

The other triads were obtained by replacing S(1)+S(5/4)+S(3/2) in the
VAR operation above by
S(1)+S(6/5)+S(3/2)             pure minor
S(1)+S(2^(4/12))+S(2^(7/12))   tempered major
S(1)+S(2^(3/12))+S(2^(7/12))   tempered minor

Dissonance functions

This demo in YouTube

In 1972 when reading the classical treatise "On the Sensations of Tone" by Helmholtz (1863)
I noticed his graphical presentation



on the consonance of various musical intervals based on practical experiments on violin.
I formulated these results as a theoretical model as follows:

Minimum points of the dissonance function

"Accuracy of the ear"        c=5     c=4     c=3     c=2

Unison                       1:1     1:1     1:1     1:1
Just minor semitone         18:17
Minor diatonic semitone     17:16
Just diatonic semitone      16:15
Septimal diatonic semitone  15:14
Lesser tridecimal 2:3-tone  14:13
Greater tridecimal 2:3-tone 13:12
Neutral second              12:11
Neutral second              11:10
Major second                10:9    10:9
Major second                 9:8     9:8
Septimal major second        8:7     8:7
                            15:13
Septimal major third         7:6     7:6
                            13:11
Just minor third             6:5     6:5     6:5
Neutral third               11:9
Major third                  5:4     5:4     5:4
Diminished major third      14:11
Septimal major third         9:7     9:7
Diminished fourth           13:10
Perfect fourth               4:3     4:3     4:3     4:3
                            15:11
Eleventh harmonic           11:8
Lesser septimal tritone      7:5     7:5
Greater septimal tritone    10:7    10:7
                            13:9
Inversion of 11th harmonic  16:11
Perfect fifth                3:2     3:2     3:2     3:2
                            17:11
Septimal minor sixth        14:9
Undecimal minor sixth       11:7    11:7
Just minor sixth             8:5     8:5
Tridecimal neutral sixth    13:8
Just major sixth             5:3     5:3     5:3
                            17:10
Septimal major sixth        12:7
Harmonic seventh             7:4     7:4     7:4
Small just minor seventh    16:9
Greater just minor seventh   9:5     9:5
Undecimal neutral seventh   11:6    11:6
                            13:7
Just major seventh          15:8
                            17:9
                            19:10
                            21:11
Octave                       2:1     2:1     2:1     2:1

Random music with Slutzky-Youle effect

This demo in YouTube

Loan payment calculator

This demo in YouTube

*
*    Loan calculator for fixed monthly payments
*
*Assume that the loan amount is K (euros), the yearly interest rate is P
*(per cent) and the loan term is T years (n=12*T months).
*The loan will be paid monthly by a constant sum x consisting of both
*the interest and amortization.
*It will be described how the monthly payment is divided into those
*two components during the entire loan period.
*Lets denote p=P/12/100 and q=1/(1+p) and thus q is the monthly discount
*factor. Then the monthly payment x is determined by requiring that
*the sum of the discounted payments must be equal to the original loan
*amount K and thus
*   K = x*q + x*q^2 + ... + x*q^n = x*q*[1 + q + q^2 + ... + q^(n-1)]
*                                 = x*q*(1-q^n)/(1-q)
*giving
*   x = K/q/(1-q^n)*(1-q).
*If during the v first months mere payment of the interest is permitted,
*the monthly payment will be
*   x = K/q/(1-q^(n-v))*(1-q)
*since the loan will be shortened only during the last n-v months.
*
*
*Lets study a case K=10000, P=5, T=10, v=6 and recall the notations
*n=12*T p=P/12/100 q=1/(1+p) .
*Our aim is to create a table with columns
*  Time     Interest     Payment         Remaining
*as a Survo data set LOAN.
*
*The columns of the table are calculated by the VAR commands
*VAR Time=ORDER TO LOAN  / Time in months 1-120
*
*Monthly=K/q/(1-q^(n-v))*(1-q) / Monthly payment (x above)
*
*Interest=if(Time<=v)then(K*p)else(Remaining[-1]*p)
*Remaining=if(Time<=v)then(K)else(Remaining[-1]-Payment)
*Payment=if(Time<=v)then(0)else(Monthly-Interest)
*VAR Interest,Payment,Remaining TO LOAN
*
*DATA LOAN,A,A+119,N,M
M  1234   1234.1234567  1234.1234567   1234567.1234567
N  Time     Interest     Payment         Remaining
A     1     41.6666667     0.0000000     10000.0000000
*     2     41.6666667     0.0000000     10000.0000000
*     3     41.6666667     0.0000000     10000.0000000
*     4     41.6666667     0.0000000     10000.0000000
*     5     41.6666667     0.0000000     10000.0000000
*     6     41.6666667     0.0000000     10000.0000000
*     7     41.6666667    68.7083351      9931.2916649
*     8     41.3803819    68.9946198      9862.2970451
*     9     41.0929044    69.2820974      9793.0149477
*    10     40.8042289    69.5707728      9723.4441749
*    11     40.5143507    69.8606510      9653.5835239
*    12     40.2232647    70.1517371      9583.4317868
*    13     39.9309658    70.4440360      9512.9877508
*    14     39.6374490    70.7375528      9442.2501980
*    15     39.3427092    71.0322926      9371.2179054
*   ---     ----------   -----------      ------------
*   115      2.7195771   107.6554247       545.0430849
*   116      2.2710129   108.1039889       436.9390960
*   117      1.8205796   108.5544222       328.3846738
*   118      1.3682695   109.0067323       219.3779415
*   119      0.9140748   109.4609270       109.9170145
*   120      0.4579876   109.9170142         0.0000003

Game of Life

This demo in YouTube

Properties of Survo in multi-tasking are shown by running four Survo sessions in separate windows simultaneously.
In three of them the Game of Life is active. All these Survo sessions are initiated in a fourth Survo session. Thus in each of the three first child processes the Game of Life is started by the main process.
The original game is here modified so that the playground is limited, the rules of transitions are generalized, and the color of each generation is selected randomly.
In this extended form the game is purely recreational without any serious purpose.
All actions take place in a Survo window and thus essentially in a MS-DOS console window.
I have programmed the Game of Life as a Survo operation LIFEGAME in 1990 and here it employs the following setup in the edit field:



There, for example, the ON and OFF specifications determine rules for `births´ and `deaths´ of elements and the LIFE specification gives the color palette. Also the borders of the upper part of the playground are visible in that picture. The startup setting in this example is always symmetric and so the game stays symmetric `forever´.
The game is started by the LIFEGAME command appearing at the end of the third edit line.

It was generated by activating the sucro of the current example twice and by moving the three 'game windows' of the second session downwards below the three first windows by the mouse. The recording was started after these activations and moves.
In this version of the demo, over 10'000 patterns will be shown in an hour.


This demo in YouTube


'Widescreen' version
For those who really like watching `Game of Life´ in this `re-creational´ form, a new version with 8 concurrent cases and lasting up to 2 hours is now available in YouTube.

Digression analysis

Digression analysis (Mixed oscillations)

This demo in YouTube

*Now the model is defined as
*MODEL M
*Y=sin(A*X+B) or Y=sin(C*X+D)
*
*and starting from 'poor' initial values
*A=8 B=12 C=15 D=0
*The DIGRES operation gives the results
*
*DIGRES SIN100,M,CUR+1
*Estimated parameters of model M:
*A=9.82352
*B=10.1049
*C=12.2938
*D=-0.154894
*n=100 rss=4.017864 R^2=0.93380 nf=671
*
*The estimates are close to the true values 10,10,12,0.
*

*
*MODEL M1
*Y=sin(A*X+B)
*
*MASK=AA--A--
*A=10 B=10 RESULTS=0
*ESTIMATE SIN100,M1,CUR+1 / IND=G,1 First group
*Estimated parameters of model M1:
*A=9.82352 (0.11347)
*B=10.1049 (0.0657281)
*n=51 rss=1.653080 R^2=0.93311 nf=39
*
*MODEL M2
*Y=sin(C*X+D)
*MASK=AA--A--
*
*C=12 D=0
*ESTIMATE SIN100,M2,CUR+1 / IND=G,2 Second group
*Estimated parameters of model M2:
*C=12.2938 (0.161625)
*D=-0.154893 (0.0904904)
*n=49 rss=2.364784 R^2=0.92972 nf=33
*

*
*DIGRES SIN100,M,CUR+1
*Estimated parameters of model M:
*A=9.82352   (0.11347)
*B=10.1049   (0.0657281)
*C=12.2938   (0.11347)
*D=-0.154894 (0.0657281)
*n=100 rss=4.017864 R^2=0.93380 nf=671
*

Plotting solutions of nonlinear Diophantine equations X^a+Y^b=cZ

This demo in YouTube

Patterns of roots in nonlinear Diophantine equations X^4+Y^4=cZ

This demo in YouTube

Grids of roots in nonlinear Diophantine equations X^4+Y^4=17*Z

This demo in YouTube


The connections between the roots of the Diophantine equation X^4+Y^4=17*Z are studied and it will be seen that they are located in the integer points of lines of type
Y=17+k*(X-17*i), k=-2, -1/2, 1/2, 2
and thus forming two slanted grids of squares. Obviously similar results are obtained in cases X^4+Y^4=c*Z when c is a prime of the form c=h*(8*n+1). This is trivially true for other values of c.

Mathematical background information


More examples of cases X^4+Y^4=c*Z where c=8n+1 is as prime or a product of such primes

For c=8*11+1=89 the grid lines have slopes k=-7/5, 5/7, -5/7, 7/5.


For c=8*9+1=73 the grid lines have slopes k=-7/3, 3/7, 7/3, -3/7.


For c=8*54+1=433 the grid lines have slopes k=-1/3, 3, 1/3, -3.


Graph of case c=697=17*41:


Combination of mod(X^4+Y^4,c)=0 and mod(X^2+Y^4,c)=0 for c=1049:

Patterns of roots in nonlinear Diophantine equations X^n+Y^n=cZ

This demo in YouTube

Solutions (X,Y) of X^n+Y^n=cZ from minimal setup

This demo in YouTube

The graphs created in the previous demos related to roots (X,Y) of Diophantine equations X^n+Y^n=cZ have a lot of symmetrical (kaleidoscopic) features.
Using the case X^32+Y^32=641*Z as an example it is shown how the graph of of roots (X,Y) for X,Y=0,1,...,2*641=1282 can be generated from a small triangular part of it by using that part or its transpose as a 'building block' 32 times.

If the same 'symmetrization' is performed by starting from a corresponding triangular part but filled randomly by points, graphs like the four ones below are obtained:




The overall structure in them is similar, but there are no accurate linear dependencies between points.

Linear clusterings of points in the graph



For finding linear clusterings numerically a sucro /DIOPH was created.

*TUTSAVE DIOPH
/
/ def Wm=W1 Wm2=W2 Wprime=W3 Wn=W4
/
*{R}
*SCRATCH {act}{home}
*DIOPH {write Wm},{write Wm2},{write Wprime},{write Wn},CUR+4{act}
*{R}
*FILE COPY DIOPH1 TO NEW DIOPH{R}
*DATA DIOPH1{R}
*p q{R}
/  The results of DIOPH are saved{R}
/  in a Survo data file DIOPH as variables p,q.{R}
*{u3}{act}
*SCRATCH {act}{home}
*VAR G:2=gcd(p,q) TO DIOPH{act}{R}
*VAR C1:1=if(p>q)then(0)else(1) TO DIOPH{act}{R}
*VAR C2:1=if(G>1)then(0)else(1) TO DIOPH{act}{R}
*VAR C3:1=C1*C2 TO DIOPH{act}{R}
*FILE COPY DIOPH TO NEW DIOPH2 / IND=C3,1{act}{R}
*VAR R:2=sqrt(p*p+q*q) TO DIOPH2{act}{R}
*FILE SORT DIOPH2 BY R TO DIOPH3{act}{R}
*FILE LOAD DIOPH3 / IND=ORDER,2,20 VARS=p,q,R{act}
*{end}

When applied to the current case mod(X^32+Y^32,641)=0   /DIOPH gives the following result:

/DIOPH 32,32,641,100
DIOPH 32,32,641,100,CUR+4                         / n_comb=533
  The results of DIOPH are saved
  in a Survo data file DIOPH as variables p,q.
VAR G:2=gcd(p,q) TO DIOPH
VAR C1:1=if(p>q)then(0)else(1) TO DIOPH
VAR C2:1=if(G>1)then(0)else(1) TO DIOPH
VAR C3:1=C1*C2 TO DIOPH
FILE COPY DIOPH TO NEW DIOPH2 / IND=C3,1
VAR R:2=sqrt(p*p+q*q) TO DIOPH2
FILE SORT DIOPH2 BY R TO DIOPH3
FILE LOAD DIOPH3 / IND=ORDER,2,20 VARS=p,q,R
DATA DIOPH3*,A,B,C
   p   q      R
   1   2      2
   1   5      5
   4   5      6
   1   8      8
   9  13     15
   5  16     16
   1  20     20
   2  25     25
   8  25     26
  17  27     31
   1  32     32
  21  31     37
  13  36     38
  19  33     38
  23  33     40
  25  32     40
   3  41     41
  12  41     42
  29  31     42

In this case the directions p/q: 1/2, 1/5, 4/5, 1/8 are the most prominent. Drawing grids of lines for them in colors 1/2 (black), 1/5 (red), 4/5 (green), 1/8 (yellow) gives the following graph:




Plotting also lines related to four next p/q directions, i.e.
9/13, 5/16, 1/20, 2/25
these first eight directions seem to cover all points (below by lines drawn in red in the first quarter of the previous graph):




Thus in the case X^32+Y^32=641*Z the set of solutions (X,Y) can be covered by 2*8 regular square grids determined by ratios
1/2, 1/5, 4/5, 1/8, 9/13, 5/16, 1/20, 2/25.

In the general case mod(X^n+Y^n,P)=0 it seems to be true that for n=2^m the maximum number of regular square grids needed to cover the set of solutions (X,Y) is 2^(m-1).

For example, for lower m values and certain P values we get by using /DIOPH:

  m    2^m   P   ratios
  2    4    17   1/2
  3    8    97   1/8   5/7
  4   16   929   11/12 4/17 7/25 14/29

General solution of mod(X^n+Y^n,P)=0 (Conjecture)


This is the fifth of 10 demos

Symmetries of roots in nonlinear Diophantine equations X^n+Y^n=cZ

This demo in YouTube

By using the Survo operation DIOPH2 is now possible to study how the roots of the Diophantine equation mod(X^n+Y^n,P)=0 can be computed in the "P square" 0<=X,Y<=P by starting from 'minimal solutions' obtained by the sucro /DIOPH and using
the basic properties:

If (X,Y) is a root then (k*X,k*Y), k=1,2,... and
(Y,X), (X,P-Y), (P-Y,X), (P-X,Y), (Y,P-X), (P-X,P-Y), (P-Y,P-X)
are also roots.

The roots (X,Y) located in the P square and related to a minimal solution (p,q) are equidistantly on parallel lines
(1) X=iPX_0+qt, Y=iPY_0+pt, i=-q,-q+1,...,p-1,p
where X_0 and Y0 are obtained from the Diophantine equation pX_0+qY_0=1 . The line for a given value of i encounters the line connecting the diagonal of the P square with end points (0,P) and (P,0) in the point determined by
t=P*(1-i*(X_0+Y_0))/(p+q).
When t is rounded to int(t)=t_0, the point of solution
X_1=iPX_0+qt_0, Y_1=iPY_0+pt_0
is located in the P square (or in its corner for i=-q and i=p). Then it is easy to determine all points of solution in the P square related to minimal solution p,q by using (1) and by the basic properties given above.

In the next demo a general procedure will be presented for determining how many minimal solutions are needed in each case. The p,q values will be processed in the order given by /DIOPH until in the set of solutions (X,Y) the next pair (p,q) already appears among those solutions.

Thus my conjecture is that the top values in the list of p,q values from /DIOPH give minimal solutions. This view is supported e.g. by the fact that a small distance sqrt(p^2+q^2) between consecutive roots implies thicker covering of roots than a big distance. Thus this principle guarantees most 'economic' approach for finding all roots.

The following graphs show these minimal solutions in two cases:



The minimal solutions are located below the red arc of a circle.

In the latter case the 32 optimal p/q ratios are
5/6, 6/7, 1/17, 11/16, 12/19, 9/22, 13/22, 2/27,
16/23, 3/29, 15/26, 21/26, 6/37, 17/33, 2/39,
9/43, 15/41, 17/40, 13/43, 9/46, 21/41, 13/46,
1/48, 25/42, 18/47, 3/53, 26/47, 32/43, 2/55,
30/49, 17/56, 31/50
in the form given by /DIOPH.

This is the sixth of 10 demos

Automatic solution of nonlinear Diophantine equations mod(X^n+Y^n,P)=0

This demo in YouTube

An algorithmic solution for determining the roots of the Diophantine equation mod(X^n+Y^n,P)=0 is presented here for n=64,P=641.

By using this technique I have solved over 100 cases with various n and P values. The results of these experiments support the following conjecture about the number of p,q pairs needed:

Assume that P is a prime number. The trivial roots (X,Y) of mod(X^n+Y^n,P)=0 are
(iP,jP), i,j=...,-2,-1,1,2,...

Let the P-1 be divisible by 2^k but not by 2^(k+1). Then non-trivial roots can appear only when n<2^k. For any even n<2^k giving also nontrivial roots, the number of p,q combinations needed for straight lines to cover the roots (X,Y) in the XY plane is
N(n,P)=ceil[gcd(n,P-1)/4].

Example: N(64,641)=16


The above N(n,P) formula holds in all cases I have tested and they are listed in
# of roots and slopes in 120 cases


Covering the points of solution by orthogonal grids with various slopes may happen in multiple ways depending on the order of p,q combinations suggested. So far the order has been determined by the gap sqrt(p^2+q^2) of consecutive points on a particular straight lines determined by p/q.
Replacing sqrt(p^2+q^2) by (p^4+q^4)^(1/4) or min(p,q) may lead to different optimal selections:

For n=32, P=641 we have

criterion          p,q ratios

sqrt(p^2+q^2)      1/2 1/5 4/5 1/8  9/13 5/16 1/20 2/25
(p^4+q^4)^(1/4)    1/2 1/5 1/8 1/20 4/5  1/50 2/25 5/16
min(p,q)           1/2 1/5 1/8 1/20 1/50 1/77 1/125 1/129

Table of p,q values for min(p,q):

///           p      q      R     X0     Y0  valid
  1           1      2      1      1      0      1
  2           1      5      1      1      0      1
  3           1      8      1      1      0      1
  4           1     20      1      1      0      1
  5           1     32      1      1      0      0
  6           1     50      1      1      0      1
  7           1     77      1      1      0      1
  8           1     80      1      1      0      0
  9           1    125      1      1      0      1
 10           1    128      1      1      0      0
 11           1    129      1      1      0      1

so that 3 suggested combinations are unnecessary.

However, in all cases the number of essential combinations is N(n,P)=8.

30 July 2018
Also when n is odd, the lines determining the locations of the the roots of mod(X^n+Y^n,P)=0 can be obtained by the principles described in this demo.
The number of different slopes of straight lines needed to cover all the roots is according to my latest experiments is simply

#slopes(n,P) = gcd(n,P-1)

when P is a prime number and this is valid also for even n values.
This is equivalent with the previous formula of N(n,P) giving the number of p,q combinations for any even n, because then each p,q combination corresponds to 4 different slopes. For odd values of n the first slope is always -1 since all (X,X)'s are roots. Also in this case when (X,Y) is a root then (Y,X) is a root.
Here is a table for six typical cases for odd n values:

n=5, P=701, gcd(n,P-1)=5
(p,q) = (-1,1) (1,4) (2,5)
               (4,1) (5,2)

n=9, P=73 , gcd(n,P-1)=9
(p,q) = (-1,1) (-1,2) (-1,4) (8,9) (4,9)
               (-2,1) (-4,1) (9,8) (9,4)

n=15, P=701, gcd(n,P-1)=5
(p,q) = (-1,1) (-8,11) (3,10)
               (-11,8) (10,3)

n=11, P=199, gcd(11,199-1)=11
(p,q) = (-1,1) (3,10) (1,11) (3,13) (11,18) (-2,7)
               (10,3) (11,1) (13,3) (18,11) (-7,2)

n=13, P=157, gcd(n,P-1)=13
(p,q) = (-1,1) (1,4) (2,7) (5,6) (3,11) (10,11) (9,13)
               (4,1) (7,2) (6,5) (11,3) (11,10) (13,9)

n=17, P=239, gcd(n,P-1)=17
(p,q) = (-1,1) (7,9) (7,10) (3,14) (9,13) (10,13) (11,14) (10,19) (-1,6)
               (9,7) (10,7) (14,3) (13,9) (13,10) (14,11) (19,10) (-6,1)

5 July 2020
When n is even the search of basic p,q combinations may be speeded up by noticing that the equation mod(X^n+Y^n,P)=0 may be replaced by

mod(X^n,P)+mod(Y^n,P)=P

giving all nontrivial roots.
For example,in the case n=64, P=641 we have

  i   p   q  mod(p^n,P) mod(q^n,P)   sum

  1   3  11  169        472          641
  2   6  11  169        472          641
  3  12  11  169        472          641
  4  15  11  169        472          641
  5   1  21    1        640          641
  6  19   9  284        357          641
  7  21   2  640          1          641
  8  21   4  640          1          641
  9  21   5  640          1          641
 10  22   3  472        169          641
 11   8  21    1        640          641
 12  13  19  357        284          641
 13  21  10  640          1          641
 14  23   9  284        357          641
 15  19  18  284        357          641
 16  11  24  472        169          641
 17  16  21    1        640          641

Then instead of scanning a lot of X,Y combinations it is sufficient to tabulate values mod(X^n,P), X=1,2,... and to find out X,Y combinations satisfying mod(X^n,P)+mod(Y^n,P)=P.

15 Nov 2020
Finding of solutions 0<=X,Y<=P can be simplified as follows:
After finding one basic solution like (1,3) in the case
mod(X^4+Y^4,41)=0 all other solutions related to this root can
be determined simply by addition and subtraction of integers.
Below the solutions for 0<=X,Y<=2P have been plotted.



In this graph the red line covers the point (1,3) and its multiples as roots.
It is easy to detect that the set of roots covered by a green line
segment is identical with the upper part of the red line segment,
since the graph consists of 2x2 identical subgraphs.
Thus the solutions on the green line are obtained by
subtracting P=41 from the Y coordinates of points on the upper part
of the red line.
By continuing the red line upwards to the point (P,3P) the last section
will be identical the third (blue) set of points ending at (P,P).
Obviously this kind of calculation can be continued for attaining
all roots in the area 0<=X,Y<=P.
The solution is completed by using symmetrical properties:
If (X,Y) is a root then
(Y,X), (X,P-Y), (P-Y,X), (P-X,Y), (Y,P-X), (P-X,P-Y), (P-Y,P-X)
are also roots.

This is the seventh of 10 demos

Stepwise display: Roots of nonlinear Diophantine equation mod(X^n+Y^n,P)=0

This demo in YouTube

For example, the last scatter diagram related to mod(X^96+Y^96,577)=0
was created in the following way:

POINTS2=1 / Save points (X,Y) of solution as a text file POINTS2.TXT
/D_SOLVE0 96,96,577,100,30

When the above sucro command was activated, it gave following output:
(corrected 31 March 2022)
..................................................
Solving Diophantine equation mod(X^96+Y^96,577)=0

Matrix PQ_X0Y0 of p,q,X0,Y0 values created!
Computing roots:
SLOPES=PQ_X0Y0 POINTS=A96_577.TXT
DIOPH3 96,96,577,577,CUR+1

Number of slopes needed is 24 .

Values of p,q,R,X0,Y0 for these 24 slopes are:

///           p      q      R     X0     Y0
  1           5      2      5      1      2
  2           6      5      8      1      1
  3           4      7      8      2      1
  4          13      4     14      1      3
  5          12      7     14      3      5
  6           1     14     14      1      0
  7           3     14     14      5      1
  8          15      2     15      1      7
  9          14      9     17      2      3
 10          16      5     17      1      3
 11          13     12     18      1      1
 12           5     17     18      7      2
 13          14     11     18      4      5
 14          18      5     19      2      7
 15           1     20     20      1      0
 16           3     20     20      7      1
 17          21      4     21      1      5
 18           9     20     22      9      4
 19          16     15     22      1      1
 20           5     22     23      9      2
 21          15     17     23      8      7
 22          20     11     23      5      9
 23          19     14     24      3      4
 24           7     23     24     10      3

..................................................

In the above run the command
DIOPH3 96,96,577,577,CUR+1
created also the sequence of solutions (X,Y) in the order
they were found and saved them in the text file POINTS2.TXT.
When in this process a solution (X,Y) was found, also its companion
solutions (Y,X),(X,P-Y),(P-Y,X),(P-X,Y),(Y,P-X),(P-X,P-Y),(P-Y,P-X)
for P=577 were saved in POINTS2.TXT.
Then the final graph emerges as a series of symmetric subgraphs.

Thereafter the graphical output was generated in this way:

FILE SAVE POINTS2.TXT TO PTS96_577 / Conversion to a data file

GPLOT PTS96_577,X1,X2
SLOW=20 / slowing down the plotting process
XSCALE=0,1154 YSCALE=0,1154

WHOME=846,2 WSIZE=920,920 HEADER=  MODE=2000,2000
POINT=0,3 WSTYLE=0 XDIV=0,1,0 YDIV=0,1,0 FRAME=0 XLABEL= YLABEL=

Colored textures by roots of nonlinear Diophantine equations mod(X^n+Y^n,P)=0

This demo in YouTube
Another set of 12 colored textures by roots
of Diophantine equations mod(X^n+Y^n,P)=0
in YouTube

Twelve colored graphs of the roots of Diophantine equations mod(X^n+Y^n,P)=0 are displayed stepwise so that points related to the directions specified by same p,q numbers have the same color.
The SLOW specification slows down the plotting speed.

In the first case related to the equation mod(X^16+Y^16,97)=0 there are four grids of points given by the sucro /D_SOLVE0:

POINTS2=1 COLOR=1
/D_SOLVE0 16,16,97,100,30

Solving Diophantine equation mod(X^16+Y^16,97)=0

SLOPES=PQ_X0Y0 POINTS=A16_97.TXT
DIOPH3 16,16,97,97,CUR+1

Number of slopes needed is 4 .

Values of p,q,R,X0,Y0,valid for these 4 slopes are:
///           p      q      R     X0     Y0  valid
  1           5      2      7      1      2      1
  2           3      5      8      2      1      1
  3           7      2      9      1      3      1
  4           7      3     10      1      2      1

The specifications POINTS2=1 and COLOR=1 take care of saving both points of solution (X,Y) and the order of the slope (C) to a text file. This text file is then converted to a Survo data file and the C values indicate the colors selected at the plotting stage by a POINT_COLOR=C specification.

The graphs separately for each slope (with straight lines connecting points) are




Thus the entire set of points (X,Y) of solution is covered by these four grids of squares.

The combination of these four graphs (without connecting lines) is the final graph



(here without a black background).


This is the ninth of 10 demos

Colored textures 2 by roots of nonlinear Diophantine equations mod(X^n+Y^n,P)=0

This demo in YouTube

The graphs of the roots of Diophantine equations mod(X^n+Y^n,P)=0 for twelve pairs of n and P values are are displayed stepwise in multiple colors. The roots related to the same p,q values (described in previous demos) have the same color.
In this way the structure of these graphs related to several overlayed slanted square grids becomes more obvious.

More information about this topic is given in the comments of the previous demo
Colored textures by roots of mod(X^n+Y^n,P)=0


Seppo Mustonen: On the roots of nonlinear Diophantic equations mod(X^n+Y^n,P)=0 (2018)

Mathematical background information

Still another set of 12 examples in YouTube

Cover picture as a contour plot

This demo in YouTube

The cover picture of the Survo User Guide (1992) is presented in various forms.


Animation of the cover picture of Survo User's guide
(Start the demo by clicking the picture below!)



This animation in YouTube


The function plotted in this demo is
f(x,y)=(x^2+y^2)*(1+a*sin(10*x)+a*sin(b*y)
for a=0(0.002)0.2 and b=10(0.05)15 simultaneously.
Then the final graph (a=0.2, b=15) is the cover picture.

The 101 frames of this movie were created by the following setup
in an edit field of Survo

*
*R=X*X+Y*Y
*X=-1,1,0.005  ZSCALING=8,0
*Y=-1,1,0.005
*TYPE=CONTOUR  SCREEN=NEG  PALETTE=VGABLUE
*XDIV=0,1000,0 YDIV=0,1000,0 FRAME=3 HEADER=
*WSIZE=1000,1000 WHOME=1000,0 WSTYLE=0
*
*TUTSAVE MOVIE
/
*{W1=0}{W2=10}{W3=0}{W4=100000}{tempo -1}{R}
+ A:
*{erase}A=R*(1+{write W1}*sin(10*X)+{write W1}*sin({write W2}*Y)){R}
*{erase}OUTFILE=P{W3=W3+1}{W4=W3+100000}{write W4}{l6}{del}{R}
*{erase}GPLOT Z(X,Y)=if(R<1)then(A)else(2047/2048){act}
*{tempo +1}{10000}{tempo -1}
*{W1=W1+0.002}{W2=W2+0.05}{R}
*GPLOT /DEL ALL{act}{home}{u3}{goto A}
*{end}

*Frames of the first half of this extension were created by the sucro MOVIE:
*
*TUTSAVE MOVIE
*{W1=0}{W2=0}{R}
+ A: {W2=W2+1}
*{erase}A=R*(1+0.2*sin({write W1}*X)+0.2*sin(1.5*{write W1}*Y)){R}
*{erase}OUTFILE=P{write W2}{R}
*{erase}GPLOT Z(X,Y)=if(R<1)then(A)else(2047/2048){act}
*{100}{R}
*>magick convert -scale x%182 P{write W2}.emf P{write W2}.png{act}{R}{30}
*>DEL P{write W2}.emf{act}{10}{R}
*{W1=W1+0.1}
*GPLOT /DEL ALL{act}{home}{u5}{goto A}
*{end}
*
*The second half was made from the first half by copying frames in
*opposite order.

Bunch of random polygons

This demo in YouTube

This is the cover picture of Guide for SURVO MM installation and setup (2002 in Finnish)
in a slightly animated form.
The graph is created by the following setup of Survo:

  1 *
  2 */ACTIVATE +
  3 +FILE MAKE POLY,4,2000,X,2
  4 *A=10
  5 +VAR X1,X2,X3,X4 TO POLY
  6 *X1=if(ORDER=1)then(0)else(X1[-1]+A*(0.5-rand(2002)))
  7 *X2=if(ORDER=1)then(0)else(X2[-1]+A*(0.5-rand(2002)))
  8 *X3=-1-8*rand(2002)
  9 *X4=if(ORDER=1)then(1)else(X41)
 10 *X41=if(X4[-1]<10)then(X4[-1]+2*rand(2002))else(0)
 11 *...........
 12 +VAR X1,X2 TO POLY
 13 *X1=if(X4>0)then(X1)else(X3)
 14 *X2=if(X4>0)then(X2)else(MISSING)
 15 *...........
 16 +GPLOT POLY,X1,X2
 17 *LINE=POLYGON,5   FRAME=0 XDIV=0,1,0 YDIV=0,1,0 HEADER= XLABEL= YLABEL=
 18 *WHOME=5,0  WSIZE=965,992 WSTYLE=0  XSCALE=-32,32 YSCALE=-43,48
 19 *FRAMES=F F=0,0,639,350,-9          LINETYPE=[line_width(0)][WHITE]
 20 *FILL(-1)=0.0,0.3,1.0,0.15
 21 *FILL(-2)=0.1,0.5,0.6,0.35
 22 *FILL(-3)=0.2,0.4,0.6,0.35
 23 *FILL(-4)=0.1,0.5,0.9,0.35
 24 *FILL(-5)=0.0,0.1,0.7,0.35
 25 *FILL(-6)=0.2,0.7,0.5,0.35
 26 *FILL(-7)=0.1,0.4,0.8,0.35
 27 *FILL(-8)=0.2,0.5,0.9,0.35
 28 *
 29 *FILL(-9)=0.01,0.01,0.05,0.30 / background
 30 *

Influence curves for the correlation coefficients 2

This demo in YouTube for -1 ≤ r ≤ 2

Influence curves of the correlation coefficient r are shown for increasing values of r when the sample size is 30, means of variables are 0 and standard deviations are 1 for -10 ≤ X,Y ≤ 10.

Formulas and background information are given in an earlier demo
Influence curves for the correlation coefficient

It is interesting to see these graphs also when r exceeds 1, but then they are statistically meaningless.
Here are some examples with very large r values:



The graphs were created by the following setup in the edit field of Survo:

 10 *   *GLOBAL*
 11 *WSIZE=1000,1020 WHOME=893,0   videomode=1000,1020
 12 *XSCALE=-10,0,10 YSCALE=-10,0,10 XDIV=0,1000,0 YDIV=0,1000,20
 13 *WSTYLE=0 FRAME=0 HEADER=
 14 *X=-10,10,0.02 Y=-10,10,0.02
 15 *
 16 *mx=0 my=0 sx=1 sy=1 n=30
 17 *  z(x,y)=abs(r*(1-w)+u*v)/w
 18 *  u=sqrt(n/(n*n-1))*(X-mx)/sx
 19 *  v=sqrt(n/(n*n-1))*(Y-my)/sy
 20 *  w=sqrt((1+u*u)*(1+v*v))
 21 *
 22 *TYPE=CONTOUR SCREEN=NEG ZSCALING=20,0
 23 *PALETTE=VGAGRAY.PAL
 24 *TEXTS=T1,T2
 25 *T1=[CourierBI(40)],___r_=_,20,993
 26 *
 27 *TUTSAVE MOVIE
 28 /
 29 *{W1=0}{W2=0}{R}
 30 + A: {tempo -1}{erase}SYSTEM videomode=1000,1020{act}{R}
 31 *{W2=W2+1}
 32 *{erase}OUTFILE=P{write W2}{R}
 33 *T2=[CourierB(40)],{write W1},170,993{R}
 34 *{erase}GPLOT z(X,Y)=abs({write W1}*(1-w)+u*v)/w{tempo +1}{act}
 35 *{wait 350}{R}
 36 *{erase}>magick convert P{write W2}.emf P{write W2}.png
 37 *{act}{R}
 38 *{wait 30}{erase}>DEL P{write W2}.emf{act}{wait 10}{R}
 39 *{W1=W1+1}GPLOT /DEL ALL{act}{home}{u6}{goto A}
 40 *{end}
 41 *
 42 *................................................
 43 *
 44 */MOVIE
 45 *SYSTEM videomode=1000,1020
 46 *OUTFILE=P1
 47 *T2=[CourierB(40)],0,170,993
 48 *GPLOT z(X,Y)=abs(0*(1-w)+u*v)/w
 49 *>magick convert P1.emf P1.png
 50 *>DEL P1.emf
 52 *GPLOT /DEL ALL
 52 *

Solving Survo puzzles by swapping numbers

This demo in YouTube

It is shown how Survo Puzzles are solved by a Javascript application.

Thirty examples with solutions Try to solve them without looking to solutions.
You may also create new problems just by giving codes of the form xyz-aaaaa where
x=# of rows, y=# of columns, z=max.number of swaps needed, and
aaaaa=serial number.

Try to solve the following Survo Puzzles by Javascript application.
353-858
443-123
363-560
343-878


Other demos about Survo puzzles:
Solving a Survo puzzle
Solving a Survo puzzle by swapping

Extended plots of roots in nonlinear Diophantine equations X^n+Y^n=cZ

This demo in YouTube

In earlier demos the roots of equations mod(X^n+Y^n,P)=0
have been plotted for non-negative X,Y values less than 2P.
Here the area has been extended to X values less than 7P
and Y values less than 4P.

Seppo Mustonen: On the roots of nonlinear Diophantic equations mod(X^n+Y^n,P)=0 (2018)

Seppo Mustonen: Diophantine equations mod(X^n+Y^n,P)=0, additional results and graphical presentations (2022)

More extended plots of roots in nonlinear Diophantine equations X^n+Y^n=cZ

This demo in YouTube

In earlier demos the roots of equations mod(X^n+Y^n,P)=0
have been plotted for non-negative X,Y values less than 2P.
Here the area has been extended to X values less than 7P
and Y values less than 4P.

In the 11th example mod(X^2+Y^2,85)=0 the parameter P=85=5*17 is not
a prime. In this case the standard slopes are 7/6,-6/7,9/2,-2/9, but
also slopes -7/6,6/7,-9/2,2/9 are needed for covering all roots.
In the graph below the covering lines are plotted for 0<=X,Y<=P .





4 Aug 2022
In fact each inner point in the graph is covered by two lines
orthogonally. Therefore all points are covered by ascending lines
with slopes 7/6, 6/7, 9/2, 2/9


Seppo Mustonen: On the roots of nonlinear Diophantic equations mod(X^n+Y^n,P)=0 (2018)

Seppo Mustonen: Diophantine equations mod(X^n+Y^n,P)=0, additional results and graphical presentations (2022)

Measurement scales in Survo

This demo in YouTube

When I started my duties as a professor of statistics in University of Helsinki in the fall of 1967 my most primary task was give a new course in mathematical statistics on the second year level. Since no corresponding teaching had not been given earlier, over 300 students participated.
One of the first topics was to introduce various scale types of statistical variables. When presenting the standard classification of variables in four categories (nominal, ordinal, interval, ratio) with permitted transformations (bijection, monotonous, linear, multiplicative), I got an idea how they are generally related to various statistical operations as follows:

Let s be a statistical operator,
e.g. s="compute the mean of observations"
and then s(x) = mean of the variable.
Then the operation s is permitted only if s(f(x))=g(s(x))
where f is any permitted transformation of the scale type of x
and g is a transformation depending on f only.
Usually we have g=f.
If s is not permitted, f(x) is a transformation of variable x.

The validity of the scale can be checked by the scale_ok C function in the library SURVO.LIB displayed below:
Excerpt (pp. 52 - 53) from the document Programming SURVO 84 in C




The statistical operations observe scale types when desired. The strictness of scale type control depends on the system parameter scale_check in the system file SURVO.APU as follows:
0 no control
1 warning when a variable with insufficient scale is selected
2 warning and interrupt of operation
Scale types can be given for variables in Survo data files only.
scale_check is 0 in the SURVO.APU file of the standard version of SURVO MM and thus scales of varibles are not checked at all.

However, when teaching statistics this checking option would be useful.

Roots of Diophantine equations mod(X^4+Y^2,P)=0

This demo in YouTube

After finding interesting structural properties for the roots
of Diophantine equations X^n+Y^n ≡ 0 (mod P) by making computational
experiments by using Survo, I tried to study equations
X^4+Y^2 ≡ 0 (mod P) in a similar way.
Since in the previous case the structures of solutions were linear,
it was natural to expect quadratic structures in this new case.
When trying to use higher degree polynomials as models
in the general ESTIMATE operation of Survo, the coefficients
in terms above degree 2 turned out to be zeros.

Diophantine equations of the form X^4+Y^2 ≡ 0 (mod P) have nontrivial
solutions for primes P of the form P=4k+1.
By making numerical experiments by means of Survo it is found that
the roots 0<=X,Y<=P for a given P are integer points on parabolas

Y=[+-(X^2-P*X)+P*i]/C(P)

where i is any integer and
C(P) is the minimal integer square root of -1 modulo P
documented as sequence A002314 in OEIS.

I have tested this result for the 16 first primes of the form P=4k+1
and below is a list of them with corresponding C(P) values.

 P     5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,149
 C(P)  2, 5, 4,12, 6, 9,23,11,27,34,22, 10, 33, 15, 37, 44

DATA XYIL
 X  Y    I  L
 0  0    2  0    I1(0,0)=0
 0 17    1  4    I1(0,17)=4
 1  4    2  0    I1(1,4)=1.8823529411765   I2(1,4)=0
 1 13    1  4    I1(1,13)=4
 2  1    1  2    I1(2,1)=2
 2 16    2  2    I1(2,16)=5.5294117647059  I2(2,16)=2
 3  2    2 -2    I1(3,2)=2.9411764705882   I2(3,2)=-2
 3 15    1  6    I1(3,15)=6
 4  4    1  4    I1(4,4)=4
 4 13    2  0    I1(4,13)=6.1176470588235  I2(4,13)=0
 5  2    1  4    I1(5,2)=4
 5 15    2  0    I1(5,15)=7.0588235294118  I2(5,15)=0
 6  8    2 -2    I1(6,8)=5.7647058823529   I2(6,8)=-2
 6  9    1  6    I1(6,9)=6
 7  8    1  6    I1(7,8)=6
 7  9    2 -2    I1(7,9)=6.2352941176471   I2(7,9)=-2
 8  1    2 -4    I1(8,1)=4.4705882352941   I2(8,1)=-4
 8 16    1  8    I1(8,16)=8
 9  1    2 -4    I1(9,1)=4.4705882352941   I2(9,1)=-4
 9 16    1  8    I1(9,16)=8
10  8    1  6    I1(10,8)=6
10  9    2 -2    I1(10,9)=6.2352941176471  I2(10,9)=-2
11  8    2 -2    I1(11,8)=5.7647058823529  I2(11,8)=-2
11  9    1  6    I1(11,9)=6
12  2    1  4    I1(12,2)=4
12 15    2  0    I1(12,15)=7.0588235294118 I2(12,15)=0
13  4    1  4    I1(13,4)=4
13 13    2  0    I1(13,13)=6.1176470588235 I2(13,13)=0
14  2    2 -2    I1(14,2)=2.9411764705882  I2(14,2)=-2
14 15    1  6    I1(14,15)=6
15  1    1  2    I1(15,1)=2
15 16    2  2    I1(15,16)=5.5294117647059 I2(15,16)=2
16  4    2  0    I1(16,4)=1.8823529411765  I2(16,4)=0
16 13    1  4    I1(16,13)=4
17  0    2  0    I1(17,0)=0
17 17    1  4    I1(17,17)=0               I2(17,17)=4

 11 *
 12 *TUTSAVE M
 13 *{tempo -1}{init}{jump 1,1,1,1}{erase}SCRATCH {home}{act}{line start}
 14 /
 15 *COLX W20{act}{line start}{erase}{tempo 2}{wait 50}{tempo -1}
 16 *{R}
 17 *   {form7} Roots of Diophantine equations mod(X^4+Y^2,P)=0 {R}
 18 *{R}
 19 *Diophantine equations of the form mod(X^4+Y^2,P)=0 have nontrivial{R}
 20 *solutions X,Y for primes P of the form P=4k+1.{R}
 21 *By making numerical experiments by means of Survo it is found that{R}
 22 *the roots X,Y for a given P are integer points on parabolas{R}
 23 *{R}
 24 *       Y=[+-(X^2-P*X)+P*i]/C(P){R}
 25 *{R}
 26 *where i is any integer and{R}
 27 *C(P) is the minimal integer square root of -1 modulo P.{R}
 28 *{R}
 29 *I have tested this result for the primes{R}
 30 *{R}
 31 *P     5,13,17,29,37,41,53,61,73   having the following C(P) values{R}
 32 *C(P)  2, 5, 4,12, 6, 9,23,11,27{R}
 33 *{R}
 34 *In this demo the case P=17 will be studied.
 35 *{tempo 2}{70}{R}{R}
 36 /
 37 *{tempo -1}{line start}
 38 *XSCALE=0,17  YSCALE=0,17 *GLOBAL*{R}
 39 *WSTYLE=0 HEADER=  XLABEL= YLABEL= XDIV=1,8,1 YDIV=1,8,1 FRAME=3{R}
 40 *WSIZE=990,990  WHOME=505,45 POINT=6,6 MODE=2000,2000{R}
 41 *....................................................{R}
 42 *{d23}{u21}{tempo 2}{10}
 43 /
 44 *The roots{tempo -1} of the equation mod(X^4+Y^2,17)=0 for X,Y<=17 are obtained{R}
 45 *by the Survo command (checking all X,Y combinations){R}
 46 *{l4}ab{r}{u2}
 47 /
 48 *DIOPH 4,2,17,17,CUR+4{tempo 2}{keys 2}{act}{keys 0}{10}
 49 /
 50 *{R}{R}
 51 *DATA K{R}
 52 *X  Y    C{20}{line start}{u2}
 53 /
 54 *GPLOT K,X,Y / WSIZE=630,630 WHOME=497,0 POINT=0,25
 55 *{keys 2}{act}{keys 0}{20}
 56 /
 57 *{home}{ins line}{ins line}{ins line}{ins line}{ins line}{ins line}
 58 *{ins line}{u5}{10}
 59 /
 60 *It is possible{tempo -1} to see{R}
 61 *that points may be{R}
 62 *located on a set{R}
 63 *of parabolas...{R}
 64 *{tempo +1}{30}
 65 /
 66 *{tempo -1}
 67 *{u}{erase}{u}{erase}{u}{erase}{u}{erase}/SOFT-PLOTCLS{act}{home}{erase}
 68 *{tempo 2}{10}
 69 /
 70 *For estimating{tempo -1} equations of these parabolas, the plot may be created{R}
 71 *so that each point is presented by its coordinates in the form x|y{R}
 72 *This is achieved as follows:{tempo 2}{30}
 73 *{R}{ref set 1}
 74 *{d4}{ins line}
 75 /
 76 *{tempo -1}{ref set 1}
 77 + A: {R}{next word}{save word W1}
 78 - if W1 '=' {sp} then goto B
 79 *{l}|{goto A}
 80 + B: {ref jump 1}{ref jump 1}{tempo +1}{10}
 81 /
 82 *I{l}{act}{act}{act}{act}{act}{act}{act}{act}{del line}{u3}{copy}a{R}
 83 *{r20}3{keys 2}{act}{keys 0}{R}
 84 *{u2}{copy}b{10}{R}
 85 *{r46}[CourierB(50)],C{keys 2}{act}{keys 0}{30}
 86 /
 87 *{home}{pre}{d}{pre}{d2}{10}
 88 /
 89 *Now following{tempo -1} points are{R}
 90 *on the same curve:{R}
 91 *DATA S:(X,Y) 7,8 10,8{R}
 92 *6,9 11,9 END{tempo +1}{R}
 93 /
 94 *{20}{tempo -1}/SOFT-PLOTCLS{tempo +1}{act}tempo -1}{home}{erase}
 95 *{tempo +1}{5}
 96 *{tempo -1}GPLOT FILE G{tempo +1}{act}{tempo -1}
 97 *{home}{erase}{tempo +1}{20}
 98 /
 99 *{tempo -1}/SOFT-PLOTCLS{act}{home}{erase}
100 *{R}
101 *{d16}{u16}{tempo 2}{15}
102 /
103 *Finding{tempo -1} the parabola going through those points by ESTIMATE operation:{R}
104 *{R}{tempo +1}
105 *MODEL M{R}
106 *Y=C*X^2+D*X+E{R}
107 *{R}
108 *ESTIMATE S,M,CUR+1 / RESULTS=0{10}{keys 2}{act}{keys 0}{20}
109 /
110 *{d2}{10}1/4{d}{l4}{10}-17/4{d}{l4}{10}3*17/2{10}{R}
111 *{R}
112 *{R}
113 *Then the formula of the parabola is{R}
114 *   {form}Y={form7}1/4*X^2-17/4*X+6*17/4{5}=(X^2-17*X+6*17)/4{5}={form}(X^2-P*X+6*P)/4 {form7}.
115 *{10}{R}
116 *Since the scatter plot{tempo -1} is symmetrical also{R}
117 *   {form}Y=(-X^2+P*X-6*P)/4{R}
118 *covers points in the graph.{tempo +1}{20}
119 /
120 *{d15}{u15}{R}{R}
121 *The complete set{tempo -1} of parabolas{R}
122 *covering all solutions{R}
123 *for X,Y=0,...,17{R}
124 *is obtained by plotting curves{R}
125 *   Y=(+X^2-P*X+i*P)/4, P=17{R}
126 *   Y=(-X^2+P*X-i*P)/4, i=-4(2)8{tempo +1}{30}{tempo -1}{R}
127 *{R}
128 *GPLOT FILE G42_17{act}{line start}{erase}{tempo +1}
129 *{end}
130 *

Roots of mod(X^n+Y^n,P)=0 by simple arithmetics

This demo in YouTube

The corresponding graph for n=2,P=5

   mod(X^2+Y^2,5)=0
    +------+
   5|*    *|
   4|  **  |
   3| *  * |
   2| *  * |
   1|  **  |
   0|*    *|
    +------+
     012345

Roots of mod(X^n+Y^n,P)=0 by simple arithmetics 2

This demo in YouTube


It is possible to determine all roots of a Diophantine equation
X^n+Y^n ≡ 0 (mod P)
when n is an even integer and P is a prime
by simple arithmetical calculations (addition and subtraction)
after the basic solutions, p,q integers close to zero are available.
In this process it is noticed that that each p,q pair gives the same amount 4P of roots.
This is evident on the basis of the proof given in comments of the preceding demo
Roots of mod(X^n+Y^n,P)=0 by simple arithmetics
In some cases the total number of roots is smaller due to overlapping.

However, the total number of solutions in the area 0<=X,Y<=P,
at least when n is a multiple of 4 and gcd(n,P-1)/4>=1,
is 4*(P-1)*N(n,P)+4 when the 4 trivial roots (0,0) (0,P) (P,0) (P,P)
are included.

N(n,P)=ceil[gcd(n,P-1)/4] is the number of starting points required,
but this has not been formally proved.
The starting points are the N(n,P) minimal solutions of mod(X^n+Y^n,P)=0
according to the distance from the origin.
It has been true in all cases presented in
# of roots and slopes in over 120 cases.

The new sucro /DI_SOLVE using a new Survo module DISOLVE finds
the correct p,q values and solutions for 0<=X,Y<=P.
The solutions are saved in a Survo data file POINTS containg values
of variables X,Y,C where C tells the relation to p,q combinations by
negative values -1,-2,...,-N(n,P).
At the end it is possible to check the result by activating
DIOPHF 32,32,577,577
which tests all X,Y combinations 0<=X,Y<=P and gives in this case
18436 as the total number of solutions.

Colored case n=100 P=401


1600 points in each of 25 p,q combinations

This graph has been created by the following means in Survo:

 11 *
 12 */DI_SOLVE 100,100,401,100
 13 *DISOLVE 100,100,401,25
 14 *
 15 *  n   p   q   points
 16 *  1   1   2   1604
 17 *  2   2   5   1600
 18 *  3   1   7   1600
 19 *  4   1   8   1600
 20 *  5   4   7   1600
 21 *  6   5   7   1600
 22 *  7   1   9   1600
 23 *  8   5   8   1600
 24 *  9   4   9   1600
 25 * 10   1  10   1600
 26 * 11   5   9   1600
 27 * 12   1  11   1600
 28 * 13   4  11   1600
 29 * 14   5  11   1600
 30 * 15   3  13   1600
 31 * 16   9  14   1600
 32 * 17   3  17   1600
 33 * 18   7  16   1600
 34 * 19  12  13   1600
 35 * 20  11  14   1600
 36 * 21   9  16   1600
 37 * 22   3  19   1600
 38 * 23   7  18   1600
 39 * 24  11  16   1600
 40 * 25  13  15   1600
 41 *
 42 *npq=25 n_total=40004 P=401
 43 *n_total=40004 is equal to 4*npq*(P-1)+4=40004
 44 *
 45 *FILE SAVE POINTS.TXT TO NEW POINTS
 46 *FILE DEL POINTS.TXT
 47 *FILE SHOW POINTS
 48 *Check results by activating:
 49 *DIOPHF 100,100,401,401
 50 *n=40004
 51 *
 52 *>REN POINTS.SVO P100_401.SVO
 53 *FILE SHOW P100_401
 54 *......................................
 55 *
 56 *WSTYLE=0 HEADER=  XLABEL= YLABEL= XDIV=0,1,0 YDIV=0,802,40 FRAME=3
 57 *WSIZE=802,842 WHOME=900,0 MODE=2000,2000 TEXTS=T
 58 *XSCALE=-1,400 YSCALE=0,400
 59 *T=[SwissB(40)][WHITE],mod(X^100+Y^100;401)_=_0___X;Y=0(1)401,100,1925
 60 *GPLOT P100_401,X,Y  / SLOW=50 WSTYLE=0
 61 *FILE SHOW P100_401
 62 *FRAME=0                      PEN=[BLACK]
 63 *POINT=0,3   POINT_COLOR=C    COLORS=[BLACK][/BLACK]
 64 *
 65 * FILL(-1)=0,0,1,0
 66 * FILL(-2)=0.5,0.5,0,0
 67 * FILL(-3)=0,0,1,0
 68 * FILL(-4)=0.5,0.5,0,0
 69 * FILL(-5)=0,0,1,0
 70 * FILL(-6)=0.5,0.5,0,0
 71 * FILL(-7)=0,0,1,0
 72 * FILL(-8)=0.5,0.5,0,0
 73 * FILL(-9)=0,0.0,0.0,0
 74 *FILL(-10)=0.5,0.5,0,0
 75 *FILL(-11)=0,0.0,0.0,0
 76 *FILL(-12)=0.5,0.5,0,0
 77 *FILL(-13)=0,0.0,0.0,0
 78 *FILL(-14)=0.5,0.5,0,0
 79 *FILL(-15)=0,0.0,0.0,0
 80 *FILL(-16)=0.5,0.5,0,0
 81 *FILL(-17)=0,0.0,0.0,0
 82 *FILL(-18)=0.5,0.5,0,0
 83 *FILL(-19)=0,0.0,0.0,0
 84 *FILL(-20)=0.5,0.5,0,0
 85 *FILL(-21)=0,0.0,0.0,0
 86 *FILL(-22)=0.5,0.5,0,0
 87 *FILL(-23)=0,0.0,0.0,0
 88 *FILL(-24)=0.5,0.5,0,0
 89 *FILL(-25)=0,0.0,0.0,0
 90 *

Roots of mod(X^n+Y^n,P)=0 by simple arithmetics 3

This demo in YouTube


The roots of Diophantine equations X^n+Y^n ≡ 0 (mod P)
can be determined by using simple arithmetics as shown in 2 previous
demos
Roots of mod(X^n+Y^n,P)=0 by simple arithmetics
Roots of mod(X^n+Y^n,P)=0 by simple arithmetics 2
In earlier demos the roots were presented from the middle of the graph
towards borders.
In this demo the roots emerge from corners towards the center
according to that simple arithmetics.


As shown in preceding demos, the roots of Diophantine equations X^n+Y^n ≡ 0 (mod P)
in the ares 0<=X,Y<=P are found when P is a prime number and
n is an even integer as follows:
At first GCD(n,P-1)/4 minimal roots X,Y are found by inspecting
points (X,Y) close to the origin (0,0).
For each of them related roots are found by projecting roots
(kX,kY), k=1,2,... to the area 0<=X,Y<=P and using the rules:
If (X,Y) is a root, then also (Y,X), (P-X,Y), (P-Y,X), (X,P-Y), (P-X,Y), (P-X,P-Y), (P-Y,P-X) are roots.
Giving the same color in the graph for all roots related
to the same minimal root emphasizes the structure of the entire solution.


Mathematical background information

More demos of the same type:

Demo 2: YouTube

Demo 3: YouTube


New summary (7 October 2022)
Seppo Mustonen: Additional results on the roots of nonlinear Diophantic equations mod(X^n+Y^n,P)=0 (2022)

Earlier summary
Seppo Mustonen: On the roots of nonlinear Diophantic equations mod(X^n+Y^n,P)=0 (2018)

Roots of mod(X^n+Y^n,P)=0 in colors

This demo in YouTube

Twelve graphs given earlier in
Automatic solution of nonlinear Diophantine equations mod(X^n+Y^n,P)=0
as monochrome static printouts are now displayed in a dynamic and colored form.

Information about creation of these graphs is given in
Technical background information

Seppo Mustonen: On the roots of nonlinear Diophantic equations mod(X^n+Y^n,P)=0 (2018)

The final forms of these 12 dynamic graphs:



Some intermediate stages of the last case mod(X^96+Y^96,577)=0
Video in YouTube



Some intermediate stages of the fifth case mod(X^24+Y^24,929)=0
Video in YouTube

Number of slopes for lines covering roots of mod(X^n+Y^n,P)=0

New demo in YouTube


When n is a multiple of 4 and P is a prime number
it is evident that the number of square grids
covering all roots of the Diophantine equation mod(X^n+Y^n,P)=0
is GCD(n,P-1)/2 (2 grids for each p,q combination)
Since each square grid depends on two parameters determining its
slope, the entire solution depends on GCD(n,P-1) slope parameters.

This interpretation can be extended to the case where n is even
and not necessarily a multiple of 4.
Then it is evident that when n is even, the number of slopes needed
for straight lines covering all roots is also GCD(n,P-1).

As an example the case n=6, P=73 where GCD(n,P-1)=6 is studied.
In this case two p,q combinations (1,3) and (3,8) are obtained.
giving 8 possible directions, but two of the (3,8) combinations
give the same solutions twice so that the roots are located
on the crossing points of the slanted grid defined by the combination
(3,8) as one can see in the graph above.
In this case the 6 slopes are 1/3, -1/3, 3/1, -3/1, 3/8, 8/3.

The following excerpt from the demo tells about these results:

Finding p,q values and number of solutions for each combination:
/DI_SOLVE 6,6,73,100
DISOLVE 6,6,73,2

  n   p   q   points
  1   1   3    292 4*(P-1)+4=292
  2   3   8    144 2*(P-1)=144

npq=2 n_total=436 P=73
FILE SAVE POINTS.TXT TO NEW POINTS / Activate!
FILE DEL POINTS.TXT                / Activate!
FILE SHOW POINTS
Check results by activating:
DIOPHF 6,6,73,73
n=436

Roots of mod(X^n+Y^n,P)=0 for odd integers n (1/2)

This demo in YouTube

As shown in the graph above the roots of the Diophantine equation
mod(X^11+Y^11,23)=0 for 0 ≤ X,Y ≤ 23 are located on straight lines with
GCD(11,23-1)=11 different slopes.
Each nontrivial root is covered by a single line.
Roots on each line are located equidistantly.
The same pattern of roots can be extended to any direction by steps n=23.

In general it is obvious that also when n is an odd integer the
number of slopes needed for straight lines to cover all roots
of Diophantine equations mod(X^n+Y^n,P)=0 when P is a prime number
is GCD(n,P-1).
This result has been indicated the earlier demo
Number of slopes for lines covering roots of mod(X^n+Y^n,P)=0
for even integers n.

In all cases the roots are the integer points on these lines.

On the basis of results obtained in these demos my general hypothesis is:

When P is a prime and n is a positive integer,
the roots of Diophantine equations mod(X^n+Y^n,P)=0
are equidistantly on straight lines with GCD(P-1,n) possible slopes
determined by roots (X,Y) closest to (0,0).
Besides trivial roots with coordinates as multiples of P,
each root is located on only one of those lines and
all integer points on those lines are roots.


So far only a limited number of odd cases have been tested.

 n   P   GCD(n,P-1)   Slopes
 3  19    3           -1 -(2,3)
 3  31    3           -1 -(1,5)
 3  37    3           -1 -(3,4)
 3  43    3           -1 -(1,6)
 3  61    3           -1 -(4,5)
 3  67    3           -1 -(2,7)
 3  73    3           -1 -(1,8)
 3  79    3           -1 -(3,7)
 3 103    3           -1 -(2,9)
 3 109    3           -1 -(5,7)
 3 127    3           -1 -(6,7)
 3 139    3           -1 -(3,10)
 5  11    5           -1 -(1,3) +(1,2)
 5  31    5           -1 -(1,2) -(1,4) or
 5  31    5           -1 -(1,2) +(3,7)
 5  41    5           -1 -(3,7) +(1,4)
 5  61    5           -1 -(2,7) +(1,3)
 5  71    5           -1 -(3,4) +(1,14)
 5 101    5           -1 -(3,7) +(1,6)
 5 131    5           -1 -(1,3) +(2,9)
 7  29    7           -1 -(2,3) -(4,5) -(1,7)
 7  43    7           -1 -(1,4) -(3,5) +(1,2)
 7  71    7           -1 -(2,3) -(4,9) +(3,7)
 7 113    7           -1 -(4,7) +(1,4) +(1,7)
 7 127    7           -1 -(1,2) -(1,4) +(7,15)
 9  13    3           -1 -(1,3)
 9  37    9           -1 -(3,4) -(5,8) -(1,7) -(1,12)
11  23   11           -1 -(1,2) -(1,3) -(2,3) -(3,4) -(1,4)
11  67   11           -1 -(3,5) -(3,8) -(5,8) +(1,3) +(1,5)
13  53   13           -1 -(2,3) -(6,7) -(4,7) -(4,9) -(7,11) -(9,11)
15  31   15           -1 -(1,2) -(4,5) -(2,5) -(1,4) -(4,7) -(5,7) -(1,10)
17 103   17           -1 -(4,7) -(1,8) -(3,10) -(5,11) -(8,9) -(5,12) +(1,3) +(4,5)
19 191   19           -1 -(1,5) -(1,6) -(5,6) -(8,9) -(8,13) -(9,13) +(3,7) +(1,11) +(5,11)
21  29    7           -1 -(2,3) -(1,7) +(1,5)
21  43   21           -1 -(2,3) -(1,4) -(2,5) -(3,5) -(1,6) -(2,7) -(3,8) -(5,7) +(1,2) +(1,3)
23  47   23           -1 -(1,2) -(1,3) -(2,3) -(1,4) -(3,4) -(1,6) +(1,5) +(2,5) +(3,5) +(4,5) +(5,6)
25  31    5           -1 -(1,2) -(1,4)
27  19    9           -1 -(2,3) -(1,4) +(1,2) +(1,3)
29  59   29           -1 -(1,3) -(1,4) -(4/3) -(5,3) -(1,5) -(1,7) -(1,9) +(2,1) +(3,2) +(5,2) +(6,1) +(7,2) +(7,6) +(8,1)

Roots of mod(X^n+Y^n,P)=0 for odd integers n (2/2)

This demo in YouTube

The roots of mod(X^n+Y^n,P)=0 for n=23, P=47 are plotted stepwise
by starting from GCD(P-1,n)=23 minimal roots
(-1,-1)
(-1,2) (-1,3) (-2,3) (-1,4) (-3,4) (-1,6) (1,5) (2,5) (3,5) (4,5) (5,6)
(-2,1) (-3,1) (-3,2) (-4,1) (-4,3) (-6,1) (5,1) (5,2) (5,3) (5,4) (6,5).
in the area 0<=X,Y<=47.

Calculations of coordinates for the multiples of these basic roots
were performed modulo P until a point in the corner of
the area 0<=X,Y<=P was reached.
The roots related to descending slopes are red and ascending ones are blue.
At the end the straight lines covering the roots are drawn.
Each root (except those in corners) is located on only one of these lines.
The number of slopes of the lines is GCD(P-1,n)=23.

In all directions P-1 nontrivial solutions were found.
It is obvious that when n is odd and GCD(P-1,n)>0,
GCD(P-1,n)*(P-1) roots for 0 < X,Y < P are found.
In this case we got GCD(P-1,n)*(P-1)+4=1062 roots.

New demo in YouTube