Jsun Yui Wong
Based on the computer program in [11], the computer program listed below seeks to solve in integers the following system of thirteen equations, including two exponential Diophantine equations from Table 1 of Perez et al. [7], 5 ^ (X(1)) + 5 ^ (X(2)) - 3 ^ (X(3)) - 7 ^ (X(4)) = 0 and 5 ^ (X(1)) + 5 ^ (X(2)) + 7 ^ (X(3)) - 17 ^ (X(4)) = 0:
X(1) * X(2) * X(3) * X(4) * X(5) * X(6) * X(7) * X(8) * X(9) * X(10) * X(11) * X(12)*X(13)*X(14)*X(15) - 1 = 0
2 * X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+X(13)+X(14)+X(15) - 16 = 0
X(1) +2 * X(2)+ X(3)+ X(4)+ X(5)+ X(6) + X(7) + X(8)+ X(9)+ X(10)+ X(11)+ X(12)+X(13)+X(14)+X(15) - 16 = 0
X(1) + X(2) + 2 * X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+X(13)+X(14)+X(15) - 16=0
X(1) + X(2) + X(3) + 2 * X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+X(13)+X(14)+X(15) - 16=0
X(1) + X(2) + X(3) + X(4) + 2 * X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+X(13)+X(14)+X(15) - 16=0
X(1) + X(2) + X(3) + X(4) + X(5) + 2 * X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+X(13)+X(14)+X(15) - 16=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + 2 * X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+X(13) +X(14) +X(15) - 16=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + 2 * X(8) + X(9) + X(10) + X(11) + X(12)+X(13)+X(14)+X(15) - 16=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+2*X(13)+X(14)+X(15) - 16 = 0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12)+X(13)+2*X(14)+X(15) - 16 = 0
5 ^ (X(1)) + 5 ^ (X(2)) - 3 ^ (X(3)) - 7 ^ (X(4)) = 0
5 ^ (X(1)) + 5 ^ (X(2)) + 7 ^ (X(3)) - 17 ^ (X(4)) = 0.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(42), A(42), L(33), K(33)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 15
94 A(KK) = FIX(RND * 5)
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 15
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15)
183 R = (1 - RND * 2) * A(B)
187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
189 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1
191 NEXT IPP
195 X(1) = -2## * X(2) - X(3) - X(4) - X(5) - X(6) - X(7) - X(8) - X(9) - X(10) - X(11) - X(12) - X(13) - X(14) - X(15) + 16##
222 N1 = 2## * X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
224 N3 = X(1) + X(2) + 2## * X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
225 N4 = X(1) + X(2) + X(3) + 2## * X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
226 N8 = X(1) + X(2) + X(3) + X(4) + 2## * X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
228 N9 = X(1) + X(2) + X(3) + X(4) + X(5) + 2## * X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
231 NZ = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + 2## * X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
232 NL = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + 2## * X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
236 REM NM = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + 2## * X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
238 REM NN = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + 2## * X(10) + X(11) + X(12) + X(13) + X(14) + X(15) - 16##
240 REM NP = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + 2## * X(11) + X(12) + X(13) + X(14) + X(15) - 16##
242 REM NQ = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + 2## * X(12) + X(13) + X(14) + X(15) - 16##
243 NR = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + 2## * X(13) + X(14) + X(15) - 16##
245 NS = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + 2## * X(14) + X(15) - 16##
246 N5 = X(1) * X(2) * X(3) * X(4) * X(5) * X(6) * X(7) * X(8) * X(9) * X(10) * X(11) * X(12) * X(13) * X(14) * X(15) - 1##
255 NNEW1 = 5 ^ (X(1)) + 5 ^ (X(2)) - 3 ^ (X(3)) - 7 ^ (X(4))
256 NNEW2 = 5 ^ (X(1)) + 5 ^ (X(2)) + 7 ^ (X(3)) - 17 ^ (X(4))
258 REM
277 P = -ABS(N1) - ABS(N3) - ABS(N4) - ABS(N5) - ABS(N8) - ABS(N9) - ABS(NZ) - ABS(NL) - ABS(NM) - ABS(NN) - ABS(NP) - ABS(NQ) - ABS(NR) - ABS(NS) - ABS(NNEW1) - ABS(NNEW2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -.3 THEN 1999
1912 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(11), A(12), A(13), A(14), A(15), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31791 is shown below:
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31928
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31890
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31875
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31791
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 [8], the wall-clock time for obtaining the output through JJJJ= -31791 was two minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Jean-Marie de Konnick, Armel Mercier, 1001 Problems in Classical Number Theory. American Mathematical Society, Providence, Rhode Island, 2007.
[2] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. www.SciRP.org/journal/am.
[3] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[4] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[5] 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.
[6] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[7] 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.
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[9] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[10] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
[11] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/testing-the-domino-method-of-general-integer-nonlinear-programming.
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