Seppo Mustonen 13 April 2021
Dynamic graphical presentations on the roots (X,Y) of
Diophantine equations mod(X^n+Y^n,P)=0
12 solutions drawn with following n and P values:
time n P
1 0:00 16 577
2 0:57 16 353
3 1:46 16 257
4 2:17 16 97
5 2:38 24 929
6 3:16 32 193
7 4:09 32 449
8 5:13 32 641
9 5:51 32 1153
10 6:28 48 577
11 7:16 8 1153
12 8:00 96 577
Interesting graphs appear when n is divisible by a high power of 2
and P is a prime number.
The greatest common divisor gcd(n,P-1) of integers n and P-1
divided by 4 gives the number of basic directions in the graph.
In each direction two square grids appear.
The size of grid squares sqrt(p*p+q*q) grows during the drawing process.
The points of solutions are equidistantly on the these grid lines
which are not drawn.
For example on the 8'th row of the table above
8 5:13 32 641
we have gcd(641-1,32)/4=8 and thus the number of square grids is 2*8=16.
Each point (X,Y) of solution fills the requirement that if both X and Y
are raised to power n=32
(i.e. they are multiplied 31 times by themselves)
and these powers are added, then the sum is divisible by P=641.
All solutions are determined through small basic solutions p,q.
In this cases those basic solutions are
p q # of solutions
1 1 2 2564 (4 corner points included)
2 1 5 2560 2564(10:factors)=2^2*641 =4*P
3 4 5 2560 Then each p,q combination produces
4 1 8 2560 4*P+4 solutions, but 4 corner points
5 9 13 2560 are recorded only once.
6 5 16 2560
7 1 20 2560
8 2 25 2560
For example X=9,Y=13 is a solution, since
9^32 is 3433683820292512484657849089281,
13^32 is 442779263776840698304313192148785281 and the sum is
9^32+13^32 442782697460660990816797849997874562
and this divided by P=641 is exactly
690768638784182512974723634942082
All multiples of basic solutions above, for example X=9*k, Y=13*k,
when k has values 1,2,3,... are also solutions
Solutions going over P=641 can be normalized below that limit
by replacing them by the remainders after division by P.
For example, the solution X=72*9=648, Y=72*13=936 is replaced
by X=648-641=7, Y=936-641=295.
Then X=7,Y=295 is a solution since
(7^32+295^32)/641=168828779853973969752013838124259484190086970488
17123087223557355427460246786
is an integer (with almost 80 digits).
In practice it is not necessary to tackle with so large integers,
since all computations can be carried out by residual arithmetic
and everything goes on by simpler calculations.
However, there is still so much computing that these tasks could not be
performed before the computer age starting on 1950'ies.
Since the basic solutions are pairs of integers on certain straight
lines, those solutions when projected to the 'plotting area'
0<=X,Y<=2P (as described above) are located on straight lines with
the same directions.
It is easy to see that when (X,Y) is a solution then also
(X,Y),(P-X,Y),(X,P-Y),(P-X,P-Y) are solutions and therefore the
solutions related to a given p,q combinations are located
equidistantly on two square grids.
This feature of solutions related to square grids justifies
coloring points on the same grid of squares with the same color
and points on different grids with other colors.
Also the background color is selected arbitrarily.
The density of points on the first grids is typically so strong that
grids seem to be created by straight lines.
The plotting speed has been slowed down by drawing each point
multiple times on the screen (sometimes over 100 times).
Then it is easier to see intermediate stages of the graph.
Some of them may be more interesting than the final graph.
Also the size of the point has a certain role.
In principle no point (except those in four corners) covers earlier
ones. On larger point sizes partial overlapping occurs.
It is interesting to see how the general outlook of the picture may
have very different shapes during the plotting process.
Graphical metamorphosis prevails!
I started these studies over three years ago in March 2018
and have produced over 10 YouTube videos about this subject.
Obviously no else have done similar things. One reason for
this is that in most cases the root patterns of these equations
are trivial multiples of X=P,Y=P and then the interesting
combinations remain undetected.
seppo.mustonen@survo.fi