Solving for Integer Solutions of Nonlinear Systems of Equations, Including Exponential Diophantine Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of 21 equations:

5 ^ X(4) + 5 ^ X(5) + 3 ^ X(6) – 3 ^ X(8)=0,
3 ^ X(5) + 5 ^ X(4) + 7 ^ X(6) – 11 ^ X(7)=0,
13 ^ X(5) + 7 ^ X(4) – 3 ^ X(8) – 5 ^ X(6)=0,
17 ^ X(4) + 7 ^ X(5) – 3 ^ X(6) – 5 ^ X(7)=0,
5 ^ X(5) + 5 ^ X(6) – 3 ^ X(8) – 7 ^ X(4)=0,
2 ^ X(5) + 7 ^ X(6) – 3 ^ X(8) – 5 ^ X(4)=-1,
-X(1) -3 + X(3) – 2 * X(5) + X(7) + X(9)=0,
(-24 – X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 – X(5) + 3 * X(6) + X(7) – 4 * X(9)) -15*X(10)=0,
-X(8) -8 + X(2) + (2 * X(4)) ^ 2 – 6 * X(6) + 2 * X(10)=0,
– 2 * X(1) – (X(2) + 3 * X(4)) ^ 3 – (5 * X(7)) ^ 2 + 6 * X(8) – X(9) + 9 * X(10)=-31,
– X(1) + 3 * X(2) – 4 * X(4) – X(6) + 6 * X(7) – X(8) + 2 * X(9)=16,
– (2 * X(1) + X(2)) ^ 2 – 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)=-27,
– 5 * X(1) – 2 * X(2) + 8 * X(4) + 3 * X(5) – 4 * X(6) – X(7) + X(9)=-23,
– 3 * X(1) – 2 * X(2) + 5 * X(3) + X(4) ^ 4 + 2 * X(5) – X(6) – 4 * X(7) + 10 * X(8) – 8 * X(9)=9,
– 3 * X(1) + (2 * X(2)) ^ 2 – 10 * X(3) + 9 * X(4) – 3 * X(5) – X(6) + 2 * X(7) + 8 * X(8) – 12 * X(9) + 5 * X(10)=25,
5 * X(1) + 10 * X(2) – 5 * X(3) + X(5) ^ 3 + 8 * X(6)=39,
6 * X(1) + X(3) – 99 * X(2) + (15 * X(6)) ^ 2=-148,
(X(1) + X(2)) ^ 2 – 7 * X(3) + 5 * X(4) + 12 * X(5) – 8 * X(6)=18,
3 * X(1) + 18 * X(3) – 5 * X(5) + 17 * X(6) = 111,
-X(1) + 5 * X(2) + 8 * X(3) – 6 * X(4) + 15 * X(5) + 10 * X(6) – 82=82,
X(2) – 5 * X(3) + 3 * X(5) – X(6) + 19=-19.

These equations, including the six exponential diophantine equations above, are based on Table 1, the 9×10 system, and the 6×6 system of Perez, Amaya, and Correa [4].
0 REM DEFDBL A-Z
1 DEFINT I, J, K

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), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 10
113 A(J44) = FIX(RND * 40)
115 NEXT J44

128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 10)
160 R = (1 – RND * 2) * A(B)

163 IF RND < .5 THEN 165 ELSE GOTO 167
165 IF RND < .5 THEN X(B) = CINT(A(B) – RND ^ 3 * R) ELSE X(B) = CINT(A(B) + RND ^ 3 * R)

166 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
168 NEXT IPP
177 X(1) = -3 + X(3) – 2 * X(5) + X(7) + X(9)
179 X(10) = (-24 – X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 – X(5) + 3 * X(6) + X(7) – 4 * X(9)) / 15
183 X(8) = -8 + X(2) + (2 * X(4)) ^ 2 – 6 * X(6) + 2 * X(10)
192 FOR J44 = 1 TO 10
193 IF X(J44) < 0 THEN 1670
194 NEXT J44
195 N(11) = 31 – 2 * X(1) – (X(2) + 3 * X(4)) ^ 3 – (5 * X(7)) ^ 2 + 6 * X(8) – X(9) + 9 * X(10)
197 N(12) = -16 – X(1) + 3 * X(2) – 4 * X(4) – X(6) + 6 * X(7) – X(8) + 2 * X(9)
199 N(13) = 27 – (2 * X(1) + X(2)) ^ 2 – 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)
201 N(14) = 23 – 5 * X(1) – 2 * X(2) + 8 * X(4) + 3 * X(5) – 4 * X(6) – X(7) + X(9)
203 N(15) = -9 – 3 * X(1) – 2 * X(2) + 5 * X(3) + X(4) ^ 4 + 2 * X(5) – X(6) – 4 * X(7) + 10 * X(8) – 8 * X(9)
205 N(16) = -25 – 3 * X(1) + (2 * X(2)) ^ 2 – 10 * X(3) + 9 * X(4) – 3 * X(5) – X(6) + 2 * X(7) + 8 * X(8) – 12 * X(9) + 5 * X(10)
209 N(17) = -39 + 5 * X(1) + 10 * X(2) – 5 * X(3) + X(5) ^ 3 + 8 * X(6)
210 N(18) = 148 + 6 * X(1) + X(3) – 99 * X(2) + (15 * X(6)) ^ 2

212 N(19) = -18 + (X(1) + X(2)) ^ 2 – 7 * X(3) + 5 * X(4) + 12 * X(5) – 8 * X(6)
213 N(20) = 5 ^ X(5) + 5 ^ X(6) – 3 ^ X(8) – 7 ^ X(4)

214 N(21) = 1 + 2 ^ X(5) + 7 ^ X(6) – 3 ^ X(8) – 5 ^ X(4)
333 REM

338 N(22) = 3 * X(1) + 18 * X(3) – 5 * X(5) + 17 * X(6) – 111

344 REM

348 N(23) = -X(1) + 5 * X(2) + 8 * X(3) – 6 * X(4) + 15 * X(5) + 10 * X(6) – 82

355 REM

359 N(24) = X(2) – 5 * X(3) + 3 * X(5) – X(6) + 19
366 N(25) = 17 ^ X(4) + 7 ^ X(5) – 3 ^ X(6) – 5 ^ X(7)

368 N(26) = 13 ^ X(5) + 7 ^ X(4) – 3 ^ X(8) – 5 ^ X(6)
369 N(27) = 3 ^ X(5) + 5 ^ X(4) + 7 ^ X(6) – 11 ^ X(7)
371 N(28) = 5 ^ X(4) + 5 ^ X(5) + 3 ^ X(6) – 3 ^ X(8)
555 P = -ABS(N(11)) – ABS(N(12)) – ABS(N(13)) – ABS(N(14)) – ABS(N(15)) – ABS(N(16)) – ABS(N(17)) – ABS(N(18)) – ABS(N(19)) – ABS(N(20)) – ABS(N(21)) – ABS(N(22)) – ABS(N(23)) – ABS(N(24)) – ABS(N(25)) – ABS(N(26)) – ABS(N(27)) – ABS(N(28))
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -1 THEN 1999

1904 PRINT A(1), A(2), A(3)
1906 PRINT A(4), A(5), A(6)
1907 PRINT A(7), A(8), A(9), A(10)
1917 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [6]. Copied by hand from the screen, the computer program’s complete output through JJJJ=32000 is shown below:

3       4       5
0       1       1
1       2       2       6
0       -19287

3       4       5
0       1       1
1       2       2       6
0       -5229

3       4       5
0       1       1
1       2       2       6
0       9242

3       4       5
0       1       1
1       2       2       6
0       15581

3       4       5
0       1       1
1       2       2       6
0       18699

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 [6], the wall-clock time through JJJJ=32000 was eight 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] 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.

[3] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.

[4] 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.

[5] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.

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

Finding Integer Solutions of Nonlinear Systems of Equations, Including Exponential Diophantine Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of 17 equations:

5 ^ X(5) + 5 ^ X(6) – 3 ^ X(8) – 7 ^ X(4)=0,
2 ^ X(5) + 7 ^ X(6) – 3 ^ X(8) – 5 ^ X(4)=-1,
-X(1) -3 + X(3) – 2 * X(5) + X(7) + X(9)=0,
(-24 – X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 – X(5) + 3 * X(6) + X(7) – 4 * X(9)) -15*X(10)=0,
-X(8) -8 + X(2) + (2 * X(4)) ^ 2 – 6 * X(6) + 2 * X(10)=0,
– 2 * X(1) – (X(2) + 3 * X(4)) ^ 3 – (5 * X(7)) ^ 2 + 6 * X(8) – X(9) + 9 * X(10)=-31,
– X(1) + 3 * X(2) – 4 * X(4) – X(6) + 6 * X(7) – X(8) + 2 * X(9)=16,
– (2 * X(1) + X(2)) ^ 2 – 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)=-27,
– 5 * X(1) – 2 * X(2) + 8 * X(4) + 3 * X(5) – 4 * X(6) – X(7) + X(9)=-23,
– 3 * X(1) – 2 * X(2) + 5 * X(3) + X(4) ^ 4 + 2 * X(5) – X(6) – 4 * X(7) + 10 * X(8) – 8 * X(9)=9,
– 3 * X(1) + (2 * X(2)) ^ 2 – 10 * X(3) + 9 * X(4) – 3 * X(5) – X(6) + 2 * X(7) + 8 * X(8) – 12 * X(9) + 5 * X(10)=25,
5 * X(1) + 10 * X(2) – 5 * X(3) + X(5) ^ 3 + 8 * X(6)=39,
6 * X(1) + X(3) – 99 * X(2) + (15 * X(6)) ^ 2=-148,
(X(1) + X(2)) ^ 2 – 7 * X(3) + 5 * X(4) + 12 * X(5) – 8 * X(6)=18,
3 * X(1) + 18 * X(3) – 5 * X(5) + 17 * X(6) = 111,
-X(1) + 5 * X(2) + 8 * X(3) – 6 * X(4) + 15 * X(5) + 10 * X(6) =82,
X(2) – 5 * X(3) + 3 * X(5) – X(6) =-19.

These equations, including the two exponential diophantine equations above, are based on Table 1, the 9×10 system, and the 6×6 system of Perez, Amaya, and Correa [3].

0 REM DEFDBL A-Z
1 DEFINT I, J, K

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), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32111

89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 10
113 A(J44) = FIX(RND * 30)

115 NEXT J44

128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 10)
160 R = (1 – RND * 2) * A(B)

163 IF RND < .5 THEN 165 ELSE GOTO 167
165 IF RND < .5 THEN X(B) = CINT(A(B) – RND ^ 3 * R) ELSE X(B) = CINT(A(B) + RND ^ 3 * R)

166 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
168 NEXT IPP
177 X(1) = -3 + X(3) – 2 * X(5) + X(7) + X(9)
179 X(10) = (-24 – X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 – X(5) + 3 * X(6) + X(7) – 4 * X(9)) / 15
183 X(8) = -8 + X(2) + (2 * X(4)) ^ 2 – 6 * X(6) + 2 * X(10)
192 FOR J44 = 1 TO 10
193 IF X(J44) < 0 THEN 1670
194 NEXT J44
195 N(11) = 31 – 2 * X(1) – (X(2) + 3 * X(4)) ^ 3 – (5 * X(7)) ^ 2 + 6 * X(8) – X(9) + 9 * X(10)
197 N(12) = -16 – X(1) + 3 * X(2) – 4 * X(4) – X(6) + 6 * X(7) – X(8) + 2 * X(9)
199 N(13) = 27 – (2 * X(1) + X(2)) ^ 2 – 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)
201 N(14) = 23 – 5 * X(1) – 2 * X(2) + 8 * X(4) + 3 * X(5) – 4 * X(6) – X(7) + X(9)
203 N(15) = -9 – 3 * X(1) – 2 * X(2) + 5 * X(3) + X(4) ^ 4 + 2 * X(5) – X(6) – 4 * X(7) + 10 * X(8) – 8 * X(9)
205 N(16) = -25 – 3 * X(1) + (2 * X(2)) ^ 2 – 10 * X(3) + 9 * X(4) – 3 * X(5) – X(6) + 2 * X(7) + 8 * X(8) – 12 * X(9) + 5 * X(10)
209 N(17) = -39 + 5 * X(1) + 10 * X(2) – 5 * X(3) + X(5) ^ 3 + 8 * X(6)
210 N(18) = 148 + 6 * X(1) + X(3) – 99 * X(2) + (15 * X(6)) ^ 2

212 N(19) = -18 + (X(1) + X(2)) ^ 2 – 7 * X(3) + 5 * X(4) + 12 * X(5) – 8 * X(6)
213 N(20) = 5 ^ X(5) + 5 ^ X(6) – 3 ^ X(8) – 7 ^ X(4)

214 N(21) = 1 + 2 ^ X(5) + 7 ^ X(6) – 3 ^ X(8) – 5 ^ X(4)

333 REM

338 N(22) = 3 * X(1) + 18 * X(3) – 5 * X(5) + 17 * X(6) – 111

344 REM

348 N(23) = -X(1) + 5 * X(2) + 8 * X(3) – 6 * X(4) + 15 * X(5) + 10 * X(6) – 82

355 REM

359 N(24) = X(2) – 5 * X(3) + 3 * X(5) – X(6) + 19
555 P = -ABS(N(11)) – ABS(N(12)) – ABS(N(13)) – ABS(N(14)) – ABS(N(15)) – ABS(N(16)) – ABS(N(17)) – ABS(N(18)) – ABS(N(19)) – ABS(N(20)) – ABS(N(21)) – ABS(N(22)) – ABS(N(23)) – ABS(N(24))
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -1 THEN 1999

1904 PRINT A(1), A(2), A(3)
1906 PRINT A(4), A(5), A(6)
1907 PRINT A(7), A(8), A(9), A(10)
1917 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [5]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-20074 is shown below:

3    4    5
0    1    1
1    2    2    6
0    -31144

3    4    5
0    1    1
1    2    2    6
0    -30420

3    4    5
0    1    1
1    2    2    6
0    -29957

3    4    5
0    1    1
1    2    2    6
0    -25944

3    4    5
0    1    1
1    2    2    6
0    -20149

3    4    5
0    1    1
1    2    2    6
0    -20074

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 [5], the wall-clock time through JJJJ=-20074 was two 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] 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.

[3] 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.

[4] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.

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

Solving in Integers Nonlinear Systems of Equations by Computer

Jsun Yui Wong

The computer program listed below is about the following quotation from http://www.math.stackexchange.com [9]:

“Do there exist integers x, y, z, w that satisfy

(x+1)^2+y^2=(x+2)^2+z^2

(x+2)^2+z^2=(x+3)^2+w^2?”
0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333), N(111), NN(111)

12 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ

16 M = -1D+317
91 FOR KK = 1 TO 4

94 A(KK) = -7000 + FIX(RND * 14001)
95 NEXT KK
128 FOR I = 1 TO 500000
129 FOR K = 1 TO 4
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 4)
182 REM IF RND < .5 THEN 183 ELSE GOTO 189
183 r = (1 – RND * 2) * A(B)

186 X(B) = A(B) + (RND ^ 3) * r

188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) – FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)

191 NEXT IPP

603 N(91) = (X(1) + 1) ^ 2 + X(2) ^ 2 – (X(1) + 2) ^ 2 – X(3) ^ 2

606 N(92) = (X(1) + 2) ^ 2 + X(3) ^ 2 – (X(1) + 3) ^ 2 – X(4) ^ 2
1433 P = -ABS(N(91)) – ABS(N(92))
1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -3 THEN 1999

1914 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s only solution through JJJJ=-30407 is shown below:

20126       -284       -201       12       0
-30407

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 [10], the wall-clock time through
JJJJ=-30407 was 20 minutes.

Acknowledgment

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

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[5] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[6] 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.
[7] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[8] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.
[9] Martin Sleziak/Puzzled417, Integer Solutions to Nonlinear System of Equations…. http://www.math.stackexchange.com/questions/2085253/integer-solutions-to-nonlinear-system-of-equations
[10] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

Solving for Integer Solutions of Nonlinear Systems of Equations

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the following nonlinear problem:

-389758 + X(2) ^ 3 + X(1) * X(6)=0,

-463176 + X(3) ^ 3 + X(2) * X(7)=0,

-136187 + X(9) ^ 3 + X(4) * X(5)=0,

-144535 + X(4) ^ 3 + X(3) * X(9)=0,

-76935 + X(8) ^ 3 + X(2) * X(6)=0,

-754559 + X(7) ^ 3 + X(1) * X(4)=0,

-61503 + X(6) ^ 3 + X(4) * X(8)=0,

-316574 + X(5) ^ 3 + X(8) * X(9)=0,

-10395 + X(1) ^ 3 + X(4) * X(5)=0,

-757393 + X(7) ^ 3 + X(7) * X(8)=0,

where 0<=X(i)<=100 and X(i) are integers for i=1, 2, 3, …, 9.

The nonlinear system above comes from Conley [6, p. 5].
0 DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333)

12 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ

16 M = -1D+317

77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91
88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128
91 FOR KK = 1 TO 9

94 A(KK) = FIX(RND * 101)
95 NEXT KK
128 FOR I = 1 TO 100000
129 FOR K = 1 TO 9
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 9)
182 REM IF RND < -.05 THEN 183 ELSE GOTO 189
183 r = (1 – RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * r

188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) – FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)

191 NEXT IPP

222 FOR J44 = 1 TO 9

228 IF X(J44) > 100 THEN 1670

233 NEXT J44

1301 N81 = -389758 + X(2) ^ 3 + X(1) * X(6)
1302 N82 = -463176 + X(3) ^ 3 + X(2) * X(7)
1303 N83 = -136187 + X(9) ^ 3 + X(4) * X(5)
1304 N84 = -144535 + X(4) ^ 3 + X(3) * X(9)

1305 N85 = -76935 + X(8) ^ 3 + X(2) * X(6)
1306 N86 = -754559 + X(7) ^ 3 + X(1) * X(4)

1307 N87 = -61503 + X(6) ^ 3 + X(4) * X(8)
1308 N88 = -316574 + X(5) ^ 3 + X(8) * X(9)

1309 N89 = -10395 + X(1) ^ 3 + X(4) * X(5)
1310 N90 = -757393 + X(7) ^ 3 + X(7) * X(8)
1445 P = -ABS(N81) – ABS(N82) – ABS(N83) – ABS(N84) – ABS(N85) – ABS(N86) – ABS(N87) – ABS(N88) – ABS(N89) – ABS(N90)
1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 9

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -4444 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), M, JJJJ

1999 NEXT JJJJ
This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s complete output through
JJJJ=-31945 is shown below:

19       73       77       52       68
39       91       42       51       0
-32000

19 73 77 52 68
39 91 42 51 0
-31999

19 73 77 52 68
39 91 42 51 0
-31998

.
.
.

19 73 77 52 68
39 91 42 51 0
-31981

19 73 77 52 68
39 91 42 51 0
-31945

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 [13, the wall-clock time through
JJJJ=-31945 was 20 seconds.

Incidentally, one can stop the computer run as soon as the first 0 (for M) appears on the screen. For the present problem, that happens at JJJJ=-32000.

Acknowledgment

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

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] William Conley, Computer Optimization Techniques, Revised Edition. Petrocelli Books, Inc., NY/Princeton, 1984.
[5] William Conley, Simulation Optimization and Correlation with Multi Stage Monte Carlo Optimization, International Journal of Systems Science, Vol.38, No. 12, Dec. 2007, pp. 1013-1019.
[6] William Conley, Ecological Optimization of Polution Control Equipment and Planning from a Simulation Perspective, International Journal of Systems Science, Vol.39, No. 1, Jan. 2008, pp. 1-7.
[7] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[8] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[11] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.
[12] Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.