Jsun Yui Wong
Based on the computer program in [16], the computer program below seeks to solve the following Diophantine equation:
X(1)^5+X(2)^5+X(3)^5+X(4)^5 = X(5)^5, which is based on the following equation taken from Lander and Parkin [5, p. 1079] and Lander and Parkin [6, p. 102]: 27^5+84^5+110^5+133^5= 144^5.
The following computer program uses qb64v1000-win [12, 15]. One notes line 88 below, which is 88 FOR JJJJ = -32000 TO 32000 STEP .5.
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 .5
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 5
112 A(J44) = 5 + (RND * 200)
113 NEXT J44
128 FOR I = 1 TO 350
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 * 8)
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 5
186 IF X(J44) > 1000 THEN X(J44) = 50
187 IF X(J44) < 10 THEN X(J44) = 50
188 NEXT J44
195 X(5) = ((X(1) ^ 5# + X(2) ^ 5# + X(3) ^ 5# + X(4) ^ 5#)) ^ .2#
215 N(7) = X(5) ^ 5# – X(1) ^ 5# – X(2) ^ 5# – X(3) ^ 5# – X(4) ^ 5#
322 PD1 = -ABS(N(7))
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 8
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -50 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT A(5), 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=32000 is shown below:
86 82 69 30
100 -43 -30476.5
69 82 86 30
100 -43 -19935.5
69 82 30 86
100 -43 -15466
133 110 27 84
144 0 -4576.5
84 27 110 133
144 0 16958
133 84 110 27
144 0 27217
Above there is no rounding by hand; it is just straight copying by hand from the screen.
At JJJJ=-4576.5, JJJJ=16958, and JJJJ=27217, M=0, which is optimal. See Lander and Parkin [5, 6].
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=32000 was two minutes and a half.
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 Brick. en.wikipedia.org/wiki/Euler_brick
[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[16] 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
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