Jsun Yui Wong
The computer program listed below seeks to solve the following 5.6.6 Diophantine Equation from Weisstein [14].
X(1) ^ 5 + X(2) ^ 5 + X(3) ^ 5 + X(4) ^ 5 + X(5) ^5 + X(6) ^ 5 = X(7) ^ 5 + X(8) ^ 5 + X(9) ^ 5 + X(10) ^ 5 + X(11) ^ 5 + X(12) ^ 5
The following computer program uses qb64v1000-win [12, 16].
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 11
112 A(J44) = 80 + FIX(RND * 500)
113 NEXT J44
128 FOR I = 1 TO 10000
129 FOR KKQQ = 1 TO 12
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 6)
140 B = 1 + FIX(RND * 11)
150 R = (1 – RND * 2) * A(B)
155 IF RND < .5 THEN 160 ELSE GOTO 167
160 X(B) = (A(B) + RND ^ 3 * R)
165 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
168 NEXT IPP
185 FOR J44 = 1 TO 12
186 IF X(J44) > 545 THEN X(J44) = 70 + FIX(RND * 700)
187 IF X(J44) < 1 THEN X(J44) = 1 + FIX(RND * 200)
188 NEXT J44
189 FOR JJ11 = 1 TO 11
190 FOR JN = JJ11 + 1 TO 12
191 IF X(JJ11) = X(JN) THEN 1670
192 NEXT JN
193 NEXT JJ11
194 IF ((X(1) ^ 5# + X(2) ^ 5# + X(3) ^ 5# + X(4) ^ 5# + X(5) ^ 5# + X(6) ^ 5# – X(7) ^ 5# – X(8) ^ 5# – X(9) ^ 5# – X(10) ^ 5# – X(11) ^ 5#)) < 0 THEN 1670
195 X(12) = ((X(1) ^ 5# + X(2) ^ 5# + X(3) ^ 5# + X(4) ^ 5# + X(5) ^ 5# + X(6) ^ 5# – X(7) ^ 5# – X(8) ^ 5# – X(9) ^ 5# – X(10) ^ 5# – -X(11) ^ 5#)) ^ .2000000000000000000000000000##
211 N(7) = X(12) ^ 5# – X(1) ^ 5# – X(2) ^ 5# – X(3) ^ 5# – X(4) ^ 5# – X(5) ^ 5# – X(6) ^ 5# + X(7) ^ 5# + X(8) ^ 5# + X(9) ^ 5# + X(10) ^ 5# + X(11) ^ 5#
322 PD1 = -ABS(N(7))
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 12
1455 A(KLX) = X(KLX)
1459 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -5 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT A(5), A(6), A(7)
1906 PRINT A(8), A(9), A(10), A(11), A(12)
1922 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [12, 16]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -22796.60000013393 is shown below:
444 463 414 279
105 10 511
89 120 141 113 445
-4 -28651.50000004873
209 525 69 270
171 404 17
11 459 134 104 503
0 -22796.60000013393
Above there is no rounding by hand; it is just straight copying by hand from the screen.
The above solution with M=0 at JJJJ= -22796.60000013393 is different from the Chen Shuwen solution presented in Weisstein [14].
One notes that 525^5=39883798828125.
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, 16], the wall-clock time for obtaining the output through JJJJ= -22796.60000013393 was three hours 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] Eric Weisstein. Diophantine Equation–5th Powers.
http://mathworld.wolfram.com/DiophantineEquation5thPowers.html
[15] Wikipedia, Euler’s sum of powers conjecture. https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture
[16] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
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