Solving in Integers a Nonlinear System of Six Simultaneous Diophantine Equations in Three Unknowns

Jsun Yui Wong

The following computer program seeks to solve in integers the nonlinear system of six simultaneous Diophantine equations in three unknowns

2 *x * (1 + y + y ^ 2)    =   3 * (1 + y ^ 4)

2 * y * (1 + z + z ^ 2)     = 3 * (1 + z ^ 4)

2 * z * (1 + x + x ^ 2)     = 3 * (1 + x ^ 4).

x+y+z   =   x^3  +y^3 +z^3

x^3  +y^3  +z^3   =  3

x+y+z   =  3.

The first three equations come from de Konnick and Mercier [1, p. 87]; the last three equations come from de Konnick and Mercier [1, p. 85];

One notes the starting vectors of line 52, which is 52 A(J44) = -100 + FIX(RND * 201).

0 REM DEFDBL A-Z
2 DEFINT I, J, X

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

    15 RANDOMIZE JJJJ

    16 M = -1D+37
    51 FOR J44 = 1 TO 3

        52 A(J44) = -100 + FIX(RND * 201)


    53 NEXT J44
    128 FOR I = 1 TO 100

        129 FOR KKQQ = 1 TO 3


            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        133 FOR IPP = 1 TO (1 + FIX(RND * 3))


            181 J = 1 + FIX(RND * 3)

            183 R = (1 - RND * 2) * A(J)
            187 X(J) = A(J) + (RND ^ 3) * R
        192 NEXT IPP

        212 X(3) = 3 - X(1) - X(2)

        360 P1 = -ABS(X(1) ^ 3 + X(2) ^ 3 + X(3) ^ 3 - 3)


        361 P2 = -ABS(X(1) + X(2) + X(3) - X(1) ^ 3 - X(2) ^ 3 - X(3) ^ 3)


        459 P3 = -ABS(2 * X(1) * (1 + X(2) + X(2) ^ 2) - 3 * (1 + X(2) ^ 4))


        460 P4 = -ABS(2 * X(2) * (1 + X(3) + X(3) ^ 2) - 3 * (1 + X(3) ^ 4))


        461 P5 = -ABS(2 * X(3) * (1 + X(1) + X(1) ^ 2) - 3 * (1 + X(1) ^ 4))


        462 IF P1 < 0 THEN P1 = P1 ELSE P1 = 0

        463 IF P2 < 0 THEN P2 = P2 ELSE P2 = 0

        464 IF P3 < 0 THEN P3 = P3 ELSE P3 = 0

        465 IF P4 < 0 THEN P4 = P4 ELSE P4 = 0


        466 IF P5 < 0 THEN P5 = P5 ELSE P5 = 0


        766 P = P1 + P2 + P3 + P3 + P4 + P5

        1111 IF P <= M THEN 1670
        1452 M = P
        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX
        1557 GOTO 128
    1670 NEXT I
    1889 IF M < 0 THEN 1999

    1947 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31953 is shown below:

1      1      1      0      -31999
1      1      1      0      -31998
1      1      1      0      -31996
1      1      1      0      -31993
1      1      1      0      -31992
1      1      1      0      -31991
1      1      1      0      -31978
1      1      1      0      -31973
1      1      1      0      -31967
1      1      1      0      -31964
1      1      1      0      -31960
1      1      1      0      -31956
1      1      1      0      -31955
1      1      1      0      -31953

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [3], the wall-clock time for obtaining the output through JJJJ= -31953 was 2 seconds, not including "Creating .EXEC file..." time.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Jean-Marie de Konnick, Armel Mercier, 1001 Problems in Classical Number Theory.  Americam Mathematical Society, Providence, Rhode Island, 2007.

[2] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[3] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[4] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.