Testing the General Nonlinear Integer Programming Solver with the 10.6.16 Diophantine Equation in Ekl

Jsun Yui Wong

The computer program listed below seeks to solve the following 10.6.16 Diophantine equation in Ekl [3, p. 1310, Table 1].

X(1) ^ 10 + X(2) ^ 10 + X(3) ^ 10 + X(4) ^ 10 + X(5) ^ 10 + X(6) ^ 10 = X(7) ^ 10 + X(8) ^ 10 + X(9) ^ 10 + X(10) ^ 10 + X(11) ^ 10 + X(12) ^ 10 + X(13) ^ 10 + X(14) ^ 10 + X(15) ^ 10 + X(16) ^ 10 + X(17) ^ 10 + X(18) ^ 10 + X(19) ^ 10 + X(20) ^ 10 + X(21) ^ 10 + X(22) ^ 10.

The following computer program uses qb64v1000-win [14, 18].

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 21
112 A(J44) = 1 + FIX(RND * 21)
113 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 22
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 21)
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 21
186 IF X(J44) > 21 THEN X(J44) = 10 + FIX(RND * 12)
187 IF X(J44) < 1 THEN X(J44) = 1 + FIX(RND * 10)
188 NEXT J44
211 IF ((X(1) ^ 10# + X(2) ^ 10# + X(3) ^ 10# + X(4) ^ 10# + X(5) ^ 10# + X(6) ^ 10# – X(7) ^ 10# – X(8) ^ 10# – X(9) ^ 10# – X(10) ^ 10# – X(11) ^ 10# – X(12) ^ 10# – X(13) ^ 10# – X(14) ^ 10# – X(15) ^ 10# – X(16) ^ 10# – X(17) ^ 10# – X(18) ^ 10# – X(19) ^ 10# – X(20) ^ 10# – X(21) ^ 10#)) < 0 THEN 1670
223 X(22) = ((X(1) ^ 10# + X(2) ^ 10# + X(3) ^ 10# + X(4) ^ 10# + X(5) ^ 10# + X(6) ^ 10# – X(7) ^ 10# – X(8) ^ 10# – X(9) ^ 10# – X(10) ^ 10# – X(11) ^ 10# – X(12) ^ 10# – X(13) ^ 10# – X(14) ^ 10# – X(15) ^ 10# – X(16) ^ 10# – X(17) ^ 10# – X(18) ^ 10# – X(19) ^ 10# – X(20) ^ 10# – X(21) ^ 10#)) ^ .100000000000000000000000000000000000000##
233 N(7) = X(22) ^ 10# – X(1) ^ 10# – X(2) ^ 10# – X(3) ^ 10# – X(4) ^ 10# – X(5) ^ 10# – X(6) ^ 10# + X(7) ^ 10# + X(8) ^ 10# + X(9) ^ 10# + X(10) ^ 10# + X(11) ^ 10# + X(12) ^ 10# + X(13) ^ 10# + X(14) ^ 10# + X(15) ^ 10# + X(16) ^ 10# + X(17) ^ 10# + X(18) ^ 10# + X(19) ^ 10# + X(20) ^ 10# + X(21) ^ 10#
322 PD1 = -ABS(N(7))
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 22
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), M, JJJJ
1924 PRINT A(9), A(10), A(11), A(12)
1944 PRINT A(13), A(14), A(15), A(16)
1954 PRINT A(17), A(18), A(19), A(20)

1964 PRINT A(21), A(22)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [14, 18]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -18549.10000019574 is shown below:

3       11       14       1
20       8       4
4       0       -18549.10000019574
16       18       7       18
7       2       10       4
4       4       4       14
17       2

Above there is no rounding by hand; it is just straight copying by hand from the screen.

The solution shown above is different from Ekl’s new solution presented in Ekl [3, p. 1310, Table 1].

One notes that 20^10=10240000000000.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [14, 18], the wall-clock time for obtaining the output through JJJJ= -18549.10000019574 was nine hours.

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] R. L. Ekl, New Results in Equal Sums of Like Powers, Mathematics of Computation, Volume 67, Number 223, July 1998, pp. 1309-1315.

[4] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[5] 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

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[10] 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.

[11] 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.

[12] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[13] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[14] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview

[15] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[16] Eric W. Weisstein. “Diophantine Equation–7th Powers.” From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/DiophantineEquation7thPowers.html

[17] Wikipedia, Euler’s sum of powers conjecture. https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture

[18] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

Testing the Nonlinear Integer Programming Solver with the 5.6.6 Diophantine Equation from Weisstein

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

Testing the Nonlinear Integer Programming Solver with Lander, Parkin, and Selfridge’s 5.1.6 Equation

Jsun Yui Wong

The computer program listed below seeks to solve the following Lander, Parkin, and Selfridge’s 5.1.6 (1967) Diophantine equation presented in Weisstein [14].

X(1) ^ 5 + X(2) ^ 5 + X(3) ^ 5 + X(4) ^ 5 + X(5) ^ 5 + X(6) ^ 5 = X(7) ^ 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 6

112 A(J44) = 1 + (RND * 100)
113 NEXT J44
128 FOR I = 1 TO 2000
129 FOR KKQQ = 1 TO 7
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)

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 6
186 IF X(J44) > 100 THEN X(J44) = 10 + FIX(RND * 100)
187 IF X(J44) < 1 THEN X(J44) = 1 + FIX(RND * 80)
188 NEXT J44

189 FOR JJ11 = 1 TO 5

190 FOR JN = JJ11 + 1 TO 6

191 IF X(JJ11) = X(JN) THEN 1670

192 NEXT JN

193 NEXT JJ11
195 X(7) = ((X(1) ^ 5# + X(2) ^ 5# + X(3) ^ 5# + X(4) ^ 5# + X(5) ^ 5# + X(6) ^ 5#)) ^ .2000000000000000000000##
211 N(7) = X(7) ^ 5# – X(1) ^ 5# – X(2) ^ 5# – X(3) ^ 5# – X(4) ^ 5# – X(5) ^ 5# – X(6) ^ 5#
322 PD1 = -ABS(N(7))
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 7
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), A(7)

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= -30230.70000002575 is shown below:

64       17       6       89
73       60       99
0       -31963.70000000053

20       96       67       4
13       19       99
0       -31541.10000000668

7       6       4       9
11       5       12
0       -30230.70000002575

Above there is no rounding by hand; it is just straight copying by hand from the screen.

These three solutions are also shown in Weisstein [14].

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= -30230.70000002575 was three 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] 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

Testing the Nonlinear Integer Programming Solver with Lander, Parkin, and Selfridge’s 5.1.7 Equation

Jsun Yui Wong

The computer program listed below seeks to solve the following Lander, Parkin, and Selfridge’s 5.1.7 (1967) 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

The following computer program uses qb64v1000-win [12, 16]. 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 7
112 A(J44) = 1 + (RND * 30)

113 NEXT J44
128 FOR I = 1 TO 2000

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) > 30 THEN X(J44) = 10 + FIX(RND * 15)

187 IF X(J44) < 1 THEN X(J44) = 1 + FIX(RND * 10)

188 NEXT J44

189 FOR JJ11 = 1 TO 6

190 FOR JN = JJ11 + 1 TO 7
191 IF X(JJ11) = X(JN) THEN 1670

192 NEXT JN

193 NEXT JJ11
195 X(8) = ((X(1) ^ 5# + X(2) ^ 5# + X(3) ^ 5# + X(4) ^ 5# + X(5) ^ 5# + X(6) ^ 5# + X(7) ^ 5#)) ^ .2000000000000000000000##
211 N(7) = X(8) ^ 5# – X(1) ^ 5# – X(2) ^ 5# – X(3) ^ 5# – X(4) ^ 5# – X(5) ^ 5# – X(6) ^ 5# – X(7) ^ 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 < 0 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4)

1905 PRINT A(5), A(6), A(7), A(8)
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= -31572.00000000623 is shown below:

7       20       18       14
1       8       15       23
0       -31963.80000000053

12       29       5       4
1       30       13       34
0       -31755.10000000356

18       15       7       20
8       14       1       23
0       -31692.80000000447

30       12       5       1
4       29       13       34
0       -31572.00000000623

Above there is no rounding by hand; it is just straight copying by hand from the screen.

The first solution shown above is also shown in Weisstein [14].

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= -31572.00000000623 was one minute.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

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[10] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

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[12] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
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[14] Eric Weisstein. Diophantine Equation–5th Powers.
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[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