A Computer Program Solving a System of Nonlinear Diophantine Equations

Jsun Yui Wong

The computer program listed below seeks to solve a system of nonlinear equations based on the following system from Kuri-Morales [6, Problem 3]:

x ^ 2 + 3 * y - z^ 3 + 10=0

x ^ 2 + y ^ 2 + z^ 2 - 6=0

x^ 3 - y ^ 2 +z  -2=0.

Here one is interested in integer solution/s only.
   

0 DEFDBL A-Z
1 DEFINT J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000


    89 RANDOMIZE JJJJ
    90 M = -3D+30
    111 FOR J44 = 1 TO 3


        112 A(J44) = -10 + FIX(RND * 20)

    115 NEXT J44

    128 FOR I = 1 TO 100


        129 FOR KKQQ = 1 TO 3
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        139 FOR IPP = 1 TO FIX(1 + RND * 3)
            140 B = 1 + FIX(RND * 3)

            150 R = (1 - RND * 2) * A(B)
            160 X(B) = (A(B) + RND ^ 3 * R)

        168 NEXT IPP

        198 X(3) = 2 - X(1) ^ 3 + X(2) ^ 2


        216 N(7) = -ABS(X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 - 6)


        217 N(8) = -ABS(X(1) ^ 2 + 3 * X(2) - X(3) ^ 3 + 10)


        322 PD1 = N(7) + N(8)


        1111 IF PD1 <= M THEN 1670
        1452 M = PD1
        1454 FOR KLX = 1 TO 4


            1455 A(KLX) = X(KLX)

        1456 NEXT KLX
        1557 REM GOTO 128
    1670 NEXT I

    1889 IF M < -5 THEN 1999


    1904 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [18]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31940 is shown below:

0      0      2      -4      -32000
0      0      2      -4      -31997
0      0      2      -4      -31995
0      0      2      -4      -31989
0      0      2      -4      -31987
0      0      2      -4      -31983
0      0      2      -4      -31978
0      0      2      -4      -31973
0      0      2      -4      -31968
0      0      2      -4      -31963
1     -1      2       0      -31960
0      0      2      -4      -31956
0      0      2      -4      -31955
0      0      2      -4      -31953
0      0      2      -4      -31948
1     -1      2       0      -31945
0      0      2      -4      -31940

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 [20], the wall-clock time for obtaining the output through JJJJ= -31940 was two seconds, not including "Creating .EXE file..." time--the total time was 9 seconds.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] I.M.M. El-Emary, M.M.Abd El-Kareem, Towards Using Genetic Algorithmfor Solving Nonlinear Equation System.  World Applied Sciences Journal 5 (3):282-289, 2008.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.388.158&rep=rep1&type=pdf

[4] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[5] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6]  Angel Fernando Kuri-Morales, Solution of Simultaneous Non-Linear Equations Using Genetic Algorithm.
www.wseas.us/e-library/conferences/brazil2002/papers/449-158.pdf

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[11] 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.

[12] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[13] Anton Schigur, Solving a System of Diophantine Equations.  Mathematics Stack Exchange. http://math.stackexchange.com/questions/398364/solving-a-system-of-diophantine-equations.

[14] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[15] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
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[16] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[17] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

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

[19] Jsun Yui Wong (2014, February 8), Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers. https://computerprogramsandresults.wordpress.com/2014/02/08/testing.