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
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