Friday, September 25, 2015

Testing the Nonlinear Integer Programming Solver with a Diophantine Equation Involving a Sum of Like Powers

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation from Reference 14 of the case of k=5 and n=5:

X(1)^5+X(2)^5+X(3)^5+X(4)^5+X(5)^5 = X(6)^5.

The following computer program uses qb64v1000-win [12, 15]. One notes line 88 below, which is 88 FOR JJJJ = -32000 TO 32000 STEP .1.

0 DEFDBL A-Z
1 DEFINT I, K, A, 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 STEP .1


    89 RANDOMIZE JJJJ
    90 M = -3D+30
    111 FOR J44 = 1 TO 6

        112 A(J44) = 5 + (RND * 150)


    113 NEXT J44
    128 FOR I = 1 TO 3000


        129 FOR KKQQ = 1 TO 8
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        139 FOR IPP = 1 TO FIX(1 + RND * 3)
            140 B = 1 + FIX(RND * 6)

            150 R = (1 - RND * 2) * A(B)
            155 IF RND < .5 THEN 160 ELSE GOTO 167
            160 X(B) = (A(B) + RND ^ 3 * R)
            164 REM IF RND<.5 THEN X(B)=(A(B)-RND*10) ELSE X(B)=(A(B) +RND*10)
            165 GOTO 168
            167 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(A(B) + 1)
        168 NEXT IPP
        169 REM GOTO 185
        171 IF X(1) = X(2) THEN 1670
        172 IF X(1) = X(3) THEN 1670
        173 IF X(2) = X(3) THEN 1670
        185 FOR J44 = 1 TO 6

            186 IF X(J44) > 1000 THEN X(J44) = 120


            187 IF X(J44) < 1 THEN X(J44) = 20


        188 NEXT J44
        195 X(6) = ((X(1) ^ 5# + X(2) ^ 5# + X(3) ^ 5# + X(4) ^ 5# + X(5) ^ 5#)) ^ .2#

        215 N(7) = X(6) ^ 5# - X(1) ^ 5# - X(2) ^ 5# - X(3) ^ 5# - X(4) ^ 5# - X(5) ^ 5#

        322 PD1 = -ABS(N(7))
        1111 IF PD1 <= M THEN 1670
        1452 M = PD1
        1454 FOR KLX = 1 TO 6

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

    1904 PRINT A(1), A(2), A(3), A(4)
    1905 PRINT A(5), A(6), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [12, 15]. Copied by hand from the screen, the computer program’s complete output through
JJJJ=-6717.200000192075 is shown below:

46      67      19      47
43      72      0        -29323.00000003896

86      92      94      38
134      144      0    -26774.30000007604

57      100      7      80
43      107      0      -18612.90000019481

46      43      67      19
47      72      0        -10640.70000020635

37      23      21      84
79      94      0        -7769.500000195903

46      47      67      19
43      72      0        -7710.200000195688

84      23      37      79
21      94      0        -7212.700000193878            

100      43      57      7
80      107      0      -6717.200000192075         .                  

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

The first solution shown above is the smallest found in 1967 by Lander, Parkin, and Selfridge, and the third solution shown above is the third smallest found in 1934 by Sastry--see Reference 14.  The fifth solution shown above is the second smallest.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [12, 15], the wall-clock time for obtaining the output through JJJJ=-6717.200000192075 was fifty minutes.

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

[4] 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

[5] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.

[6] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[7] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[8] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[9] 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.

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

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

[12] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview

[13] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[14] Wikipedia, Euler's sum of powers conjecture.  https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture

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

No comments:

Post a Comment