Jsun Yui Wong
Based on the code in [15], 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.
0 DEFDBL A-Z
2 DEFINT J, X
3 DIM N(32999), A(32999), H(32999), L(32999), U(32999), X(32999), D(32999), P(32999), PS(32999), J(32999)
12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ
16 M = -1D+37
41 FOR J44 = 1 TO 5
42 A(J44) = 20 + FIX(RND * 100)
43 NEXT J44
128 FOR I = 1 TO 25000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 J = 1 + FIX(RND * 5)
182 IF RND < .5 THEN 183 ELSE GOTO 191
183 R = (1 - RND * 2) * A(J)
185 IF R > 30000 THEN 1670
189 X(J) = A(J) + FIX(RND * R)
190 GOTO 192
191 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1
192 NEXT IPP
233 FOR J44 = 1 TO 5
237 IF X(J44) < 20 THEN X(J44) = A(J44)
244 NEXT J44
265 X(5) = (X(3) ^ 5 + X(4) ^ 5 + X(1) ^ 5 + X(2) ^ 5) ^ (1 / 5)
441 P(12500) = -ABS(X(5) ^ 5 - X(1) ^ 5 - X(2) ^ 5 - X(3) ^ 5 - X(4) ^ 5)
452 IF P(12500) < 0 THEN P(12500) = P(12500) ELSE P(12500) = 0
595 P = P(12500)
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -40 THEN 1999
1947 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s complete output through JJJJ=32000 is shown below:
133 27 84 110 144
0 -30617
133 84 27 110 144
0 -25611
110 27 84 133 144
0 -14526
133 27 84 110 144
0 -10921
110 84 133 27 144
0 -75
27 84 133 110 144
0 5583
110 27 133 84 144
0 9250
84 27 110 133 144
0 15844
84 133 110 27 144
0 17440
27 110 133 84 144
0 18646
27 84 110 133 144
0 25383
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 [13], the wall-clock time for obtaining the output through JJJJ=32000 was 70 minutes.
About two centuries after Euler made his conjecture, Lander and Parkin [5] published in 1966 their counterexample
27^5+84^5+110^5+133^5= 144^5, [Gardner [4], page 16].
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). 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. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.
[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[14] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html.
[15] Jsun Yui Wong (2014, February 10), Testing the Nonlinear Integer Programming Solver with Lander and Parkin's Counterexample to Euler's Conjecture on Sums of Like Powers, Third Edition. http://myblogsubstance.typepad.com/substance/2014/02/testing-the-nonlinear-integer-programming-solver-with-lander-and-parkins-counterexample-to-eulers-co-1.html.