The computer program listed below seeks to solve the following 7.4.4 Diophantine equation from Weisstein [15].
X(1) ^ 7 + X(2) ^ 7 + X(3) ^ 7 + X(4) ^ 7 = X(5) ^ 7 + X(6) ^ 7 + X(7) ^ 7 + X(8) ^ 7
Weisstein [15]: "Guy (1994, p. 140) asked if a 7.4.4 equation exists." See Guy [5].
The following computer program uses qb64v1000-win [13, 17].
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 7
112 A(J44) = 10 + (RND * 200)
113 NEXT J44
128 FOR I = 1 TO 15000
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 * 7)
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 7
186 IF X(J44) > 155 THEN X(J44) = 50 + FIX(RND * 200)
187 IF X(J44) < 1 THEN X(J44) = 1 + FIX(RND * 20)
188 NEXT J44
189 FOR JJ11 = 1 TO 7
190 FOR JN = JJ11 + 1 TO 8
191 IF X(JJ11) = X(JN) THEN 1670
192 NEXT JN
193 NEXT JJ11
194 IF ((X(1) ^ 7# + X(2) ^ 7# + X(3) ^ 7# + X(4) ^ 7# - X(5) ^ 7# - X(6) ^ 7# - X(7) ^ 7#)) < 0 THEN 1670
195 X(8) = ((X(1) ^ 7# + X(2) ^ 7# + X(3) ^ 7# + X(4) ^ 7# - X(5) ^ 7# - X(6) ^ 7# - X(7) ^ 7#)) ^ .1428571428571428571428571428571428571428##
211 N(7) = X(8) ^ 7# - X(1) ^ 7# - X(2) ^ 7# - X(3) ^ 7# - X(4) ^ 7# + X(5) ^ 7# + X(6) ^ 7# + X(7) ^ 7#
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)
1459 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -4990 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT A(5), A(6), A(7)
1906 PRINT A(8), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [13, 17]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -31586.80000000601 is shown below:
15 140 9 113
66 144 30
16 -1785 -31707.60000000426
16 109 32 74
107 65 84
17 -4368 -31649.5000000051
129 90 15 146
10 123 149
14 0 -31586.80000000601
Above there is no rounding by hand; it is just straight copying by hand from the screen.
The solution shown above at JJJJ= -31586.80000000601 with M=0 is the same as one of the three solutions presented in Weisstein [15].
One notes that 149^7=1630436461403549.
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, 17], the wall-clock time for obtaining the output through
JJJJ= -31586.80000000601 was ten 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] R. K. Guy, "Sums of Like Powers. Euler's Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-144, 1994.
[6] 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.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021
[12] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.
[13] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview
[14] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.
[15] Eric W. Weisstein. "Diophantine Equation--7th Powers." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/DiophantineEquation7thPowers.html
[16] Wikipedia, Euler's sum of powers conjecture. https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture
[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
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