A Nonlinear Integer Programming Code/Software/Solver Applied to Solving a Nonlinear System of 32500 Simultaneous Diophantine Equations Based on the Brown Almost Linear Function

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear system of 32500 Diophantine equations:

32500
x(i) + sigma x(j) – (32500+1) = 0, for i = 1, 2, 3,…, 32499,
j=1

32500
pi      x(j) -1 = 0.
j=1

This present system is based on the Brown almost linear function in La Cruz, Marinez, and Raydan [3, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Han and Han [2, p. 227, Example 3]; http://www.SciRP.org/journal/am.

The starting vectors are shown in line 42, which is 42 A(J44) = -3 + FIX(RND * 7).

0 REM DEFDBL A-Z
2 DEFINT J, X

3 DIM B(30999), N(32999), A(32999), H(32999), L(32999), U(32999), X(32999), D(32999), P(32999), PS(32999), J(32999)

12 FOR JJJJ = -32000 TO 32000

15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 32500
42 A(J44) = -3 + FIX(RND * 7)

43 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 32500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 32503)
182 IF RND < .5 THEN 183 ELSE GOTO 191
183 R = (1 – RND * 2) * A(J)

184 IF R = 0 THEN 186 ELSE GOTO 189
185 REM

186 IF RND < .5 THEN X(J) = -1 ELSE X(J) = 1
187 REM
188 GOTO 192

189 X(J) = A(J) + (RND) * R
190 GOTO 192
191 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
192 NEXT IPP

251 SU = 0
254 FOR J44 = 1 TO 32499
258 SU = SU + X(J44)
266 NEXT J44

311 X(32500) = -X(1) – SU + (32500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 32500
355 PR = PR * X(J45)

359 NEXT J45
422 FOR J41 = 2 TO 32499
439 P(J41) = -ABS(X(J41) + SU + X(32500) – (32500 + 1))
427 NEXT J41
441 P(32500) = -ABS(PR – 1)
451 FOR J77 = 2 TO 32500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0

454 NEXT J77
577 SP = 0

578 FOR J99 = 2 TO 32500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 32500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 REM IF M < -99 THEN 1999

1947 PRINT A(1), A(2), A(3), A(4), A(32497), A(32498), A(32499), A(32500), 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=-31998 is shown below:

-1       0       2       -2       1
-3       -2       -32528       -2.129788E+09
-32000

1       1       1       1       1
1       1       1       0       -31999

1       1       1       1       1
1       1       6       -6       -31998

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

Thus, through JJJJ=-31998, M=0 was obtained at JJJJ=-31999. Of the 32500 A’s, only the 8 A’s of line 1947 are shown above.

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= -31998 was four hours and a half.

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. http://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] 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.

A Nonlinear Integer Programming Code/Software/Solver Applied to Solving a Nonlinear System of 30500 Simultaneous Diophantine Equations Based on the Brown Almost Linear Function

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear system of 30500 Diophantine equations:

30500
x(i) + sigma x(j) – (30500+1) = 0, for i = 1, 2, 3,…, 30499,
j=1

30500
pi     x(j) -1 = 0.
j=1

This present system is based on the Brown almost linear function in La Cruz, Marinez, and Raydan [3, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Han and Han [2, p. 227, Example 3]; http://www.SciRP.org/journal/am.

The starting vectors are shown in line 42, which is 42 A(J44) = -3 + FIX(RND * 7).

0 REM DEFDBL A-Z
2 DEFINT J, X

3 DIM B(30999), N(30999), A(30999), H(30999), L(30999), U(30999), X(30999), D(30999), P(30999), PS(30999), J(30999)

12 FOR JJJJ = -32000 TO 32000

15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 30500
42 A(J44) = -3 + FIX(RND * 7)
43 NEXT J44
128 FOR I = 1 TO 300000
129 FOR KKQQ = 1 TO 30500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 30503)
182 IF RND < .5 THEN 183 ELSE GOTO 191

183 R = (1 – RND * 2) * A(J)
187 REM IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1

189 X(J) = A(J) + (RND ^ 3) * R

190 GOTO 192
191 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
192 NEXT IPP

251 SU = 0
254 FOR J44 = 1 TO 30499
258 SU = SU + X(J44)
266 NEXT J44

311 X(30500) = -X(1) – SU + (30500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 30500
355 PR = PR * X(J45)

359 NEXT J45
422 FOR J41 = 2 TO 30499
439 P(J41) = -ABS(X(J41) + SU + X(30500) – (30500 + 1))
427 NEXT J41
441 P(30500) = -ABS(PR – 1)
451 FOR J77 = 2 TO 30500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0

454 NEXT J77
577 SP = 0

578 FOR J99 = 2 TO 30500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 30500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 REM IF M < -99 THEN 1999

1947 PRINT A(1), A(2), A(3), A(4), A(30497), A(30498), A(30499), A(30500), 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=-31999 is shown below:

0       0       0       0       0
0       0       30501       -1       -32000

1       1       1       1       1
1       1       1       0       -31999

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

Thus, through JJJJ=-31999, M=0 was obtained at JJJJ=-31999. Of the 30500 A’s, only the 8 A’s of line 1947 are shown above.

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= -31999 was five hours.

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. http://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] 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.

A Nonlinear Integer Programming Code/Software/Solver Applied to Solving a Nonlinear System of 20500 Simultaneous Diophantine Equations Based on the Brown Almost Linear Function

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear system of 20500 Diophantine equations:

20500
x(i) + sigma x(j) – (20500+1) = 0, for i = 1, 2, 3,…, 20499,
j=1

20500
pi    x(j) -1 = 0.
j=1

This present system is based on the Brown almost linear function in La Cruz, Marinez, and Raydan [3, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Han and Han [2, p. 227, Example 3]; http://www.SciRP.org/journal/am.

The starting vectors are shown in line 42, which is 42 A(J44) = -1 + FIX(RND * 3).

0 REM DEFDBL A-Z
2 DEFINT J, X

3 DIM B(20999), N(20999), A(20999), H(20999), L(20999), U(20999), X(20999), D(20999), P(20999), PS(20999), J(20999)

12 FOR JJJJ = -32000 TO 32000

15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 20500
42 A(J44) = -1 + FIX(RND * 3)

43 NEXT J44
128 FOR I = 1 TO 100000

129 FOR KKQQ = 1 TO 20500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 20503)

183 REM R = (1 – RND * 2) * A(J)

187 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
189 REM X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP

251 SU = 0
254 FOR J44 = 1 TO 20499
258 SU = SU + X(J44)
266 NEXT J44

311 X(20500) = -X(1) – SU + (20500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 20500
355 PR = PR * X(J45)

359 NEXT J45
422 FOR J41 = 2 TO 20499
439 P(J41) = -ABS(X(J41) + SU + X(20500) – (20500 + 1))
427 NEXT J41
441 P(20500) = -ABS(PR – 1)
451 FOR J77 = 2 TO 20500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0

454 NEXT J77
577 SP = 0

578 FOR J99 = 2 TO 20500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 20500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 REM IF M < -99 THEN 1999

1947 PRINT A(1), A(2), A(3), A(4), A(20497), A(20498), A(20499), A(20500), 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=-31999 is shown below:

0       0       0       0       0
0       0       20501       -1       -32000

1       1       1       1       1
1       1       1       0       -31999

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

Thus, through JJJJ=-31999, M=0 was obtained at JJJJ=-31999. Of the 20500 A’s, only the 8 A’s of line 1947 are shown above.

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= -31999 was two hours.

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. http://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] 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.

A Nonlinear Integer Programming Code/Software/Solver Applied to Solving a Nonlinear System of 9500 Simultaneous Diophantine Equations Based on the Brown Almost Linear Function

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear system of 9500 Diophantine equations:

9500
x(i) + sigma x(j) – (9500+1) = 0, for i = 1, 2, 3,…, 9499,
j=1

9500
pi x(j) -1 = 0.
j=1

This present system is based on the Brown almost linear function in La Cruz, Marinez, and Raydan [3, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Han and Han [2, p. 227, Example 3]; http://www.SciRP.org/journal/am.

The starting vectors are shown in line 42, which is 42 A(J44) = -3 + FIX(RND * 7).

0 REM DEFDBL A-Z
2 DEFINT J, X

3 DIM B(9999), N(9999), A(9999), H(9999), L(9999), U(9999), X(9999), D(9999), P(9999), PS(9999), J(9999)

12 FOR JJJJ = -32000 TO 32000

15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 9500
42 A(J44) = -3 + FIX(RND * 7)
43 NEXT J44
128 FOR I = 1 TO 50000

129 FOR KKQQ = 1 TO 9500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 9500)

183 REM R = (1 – RND * 2) * A(J)

187 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
189 REM X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP

251 SU = 0
254 FOR J44 = 1 TO 9499
258 SU = SU + X(J44)
266 NEXT J44

311 X(9500) = -X(1) – SU + (9500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 9500
355 PR = PR * X(J45)

359 NEXT J45
422 FOR J41 = 2 TO 9499
439 P(J41) = -ABS(X(J41) + SU + X(9500) – (9500 + 1))
427 NEXT J41
441 P(9500) = -ABS(PR – 1)
451 FOR J77 = 2 TO 9500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0

454 NEXT J77
577 SP = 0

578 FOR J99 = 2 TO 9500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 9500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 REM IF M < -99 THEN 1999

1947 PRINT A(1), A(2), A(3), A(4), A(7497), A(9498), A(9499), A(9500), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [7]. Copied by hand from the screen, the computer program’s output through JJJJ=-31962 is summarized below:

0 0 0 0 0
0 0 9501 -1 -32000

-1 -1 -1 -1 -1
-1 -1 19000 -2 -31999

0 0 0 0 0
0 0 9504 -4 -31998

1 1 1 1 1
1 1 2 -2 -31997
.
.
.

1 1 1 1 1
1 1 1 0 -31979
.
.
.

1 1 1 1 1
1 1 1 0 -31974
.
.
.

1 1 1 1 1
1 1 1 0 -31967
.
.
.

1 1 1 1 1
1 1 1 0 -31964

1 1 1 1 1
1 1 1 0 -31963

0 0 0 0 0
0 0 9500 -2 -31962

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

In summary, M=0 at JJJJ=-31979, JJJJ=-31974, JJJJ=-31967, JJJJ=-31964, and JJJJ=-31963. Of the 9500 A’s, only the 8 A’s of line 1947 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [7], the wall-clock time for obtaining the output through JJJJ= -31962 was ten hours.

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. http://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] 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.

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

[8] 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.

A Nonlinear Integer Programming Code/Software/Solver Applied to Solving a Nonlinear System of 4500 Simultaneous Diophantine Equations Based on the Brown Function

Jsun Yui Wong

The computer program lissted below seeks to solve the following  nonlinear system of 4500 Diophantine equations:

4500
x(i) + sigma x(j) – (4500+1) = 0, for i = 1, 2, 3,…, 4499,
j=1

4500
pi    x(j) -1 = 0.
j=1

This present system is based on Brown’s almost linear function in La Cruz, Marinez, and Raydan [2, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

The starting vectors are shown in line 42, which is 42 A(J44) = -5 + FIX(RND * 11).

0 REM DEFDBL A-Z
2 DEFINT I, J, X

3 DIM B(9999), N(9999), A(9999), H(9999), L(9999), U(9999), X(9999), D(9999), P(9999), PS(9999), J(9999)

12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 4500
42 A(J44) = -5 + FIX(RND * 11)
43 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 4500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 4500)

183 REM R = (1 – RND * 2) * A(J)

187 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
189 REM X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP

251 SU = 0
254 FOR J44 = 1 TO 4499
258 SU = SU + X(J44)
266 NEXT J44

311 X(4500) = -X(1) – SU + (4500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 4500
355 PR = PR * X(J45)

359 NEXT J45
422 FOR J41 = 2 TO 4499
439 P(J41) = -ABS(X(J41) + SU + X(4500) – (4500 + 1))
427 NEXT J41
441 P(4500) = -ABS(PR – 1)

451 FOR J77 = 2 TO 4500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0

454 NEXT J77
577 SP = 0

578 FOR J99 = 2 TO 4500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 REM IF M < -99 THEN 1999

1947 PRINT A(1), A(2), A(3), A(4), A(4497), A(4498), A(4499), A(4500), 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=-31989 is shown below:

0    0    0    0    0
0   0    4503    -3    -32000

2    2    2    2    2
2    2    -4494    -5    -31999

1    1    1    1    1
1    1    2    -2    -31998

-1    -1    -1    -1    -1
-1    -1    9000    -2    -31997

1    1    1    1    1
1    1    1       0       -31996

-2    -2    -2    -2    -2
-2    -2    13488    -13    -31995

3    3    3    3    3
3    3    -8988    -11    -31994

-2    -2    -2    -2    -2
-2    -2    13490    -11    -31993

1    1    1    1    1
1    1    4    -4    -31992

-2    -2    -2    -2    -2
-2    -2       13486    -15    -31991

1    1    1    1    1
1    1    6    -6       -31990

1    1    1    1    1
1    1    1       0       -31989

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

Above at JJJJ=-31996 and at JJJJ= -31989, M=0. Of the 4500 A’s, only the eight A’s of line 1947 are shown above.

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 for obtaining the output through JJJJ= -31989 was one hour.

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] 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.

[3] 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.

[4] 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.

[5] 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.

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

[7] 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.

Simultaneously Solving in Integers a Nonlinear System of 4500 Diophantine Equations in 4500 Unknowns

Jsun Yui Wong

The following computer program seeks to solve the following nonlinear system of 4500 Diophantine equations:

4500
x(i) + sigma x(j) – (4500+1) = 0, for i = 1, 2, 3,…, 4499,
j=1

4500
pi x(j) -1 = 0.
j=1

This present system is based on Brown’s almost linear function in La Cruz, Marinez, and Raydan [2, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

While line 42 of the preceding paper is 42 A(J44) = -5 + FIX(RND * 11), here line 42 is 42 A(J44) = -15 + FIX(RND * 31); the latter has a longer range.

0 REM DEFDBL A-Z
2 DEFINT I, J, X

3 DIM B(9999), N(9999), A(9999), H(9999), L(9999), U(9999), X(9999), D(9999), P(9999), PS(9999), J(9999)

12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 4500
42 A(J44) = -15 + FIX(RND * 31)
43 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 4500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 4500)

183 REM R = (1 – RND * 2) * A(J)

187 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
189 REM X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP

251 SU = 0
254 FOR J44 = 1 TO 4499
258 SU = SU + X(J44)
266 NEXT J44

311 X(4500) = -X(1) – SU + (4500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 4500
355 PR = PR * X(J45)

359 NEXT J45
422 FOR J41 = 2 TO 4499
439 P(J41) = -ABS(X(J41) + SU + X(4500) – (4500 + 1))
427 NEXT J41
441 P(4500) = -ABS(PR – 1)

451 FOR J77 = 2 TO 4500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0

454 NEXT J77
577 SP = 0

578 FOR J99 = 2 TO 4500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 REM IF M < -99 THEN 1999

1947 PRINT A(1), A(2), A(3), A(4), A(4497), A(4498), A(4499), A(4500), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [6]. Copied by hand from the screen, the computer program’s output through JJJJ=-31962 is summarized below:

0 0 0 0 0
0 0 4501 -1 -32000

-6 -6 -6 -6 -6
-6 -6 31488 -13 -31999

4 4 4 4 4
4 4 -13493 -6 -31998

2 2 2 2 2
2 2 -4491 -8 -31997

7 7 7 7 7
7 7 -26991 -8 -31996

-1 -1 -1 -1 -1
-1 -1 9000 -2 -31995
.
.
.
1 1 1 1 1
1 1 1 0 -31962

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

Above at JJJJ=-31962, M=0. Of the 4500 unknowns, only the eight A’s of line 1947 are shown above.

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 for obtaining the output through
JJJJ= -31962 was six hours.

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] 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.

[3] 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.

[4] 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.

[5] 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.

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

[7] 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.

Simultaneously Solving in Integers a Nonlinear System of 6500 Diophantine Equations in 6500 Unknowns

Jsun Yui Wong

The following computer program seeks to solve the following nonlinear system of 6500 Diophantine equations:

6500
x(i) + sigma x(j) – (6500+1) = 0, for i = 1, 2, 3,…, 6499,
j=1

6500
pi   x(j) -1 = 0.
j=1

This present system is based on Brown’s almost linear function in La Cruz, Marinez, and Raydan [2, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

One notes the starting vectors of line 42, which is 42 A(J44) = -5 + FIX(RND * 11).

0 REM DEFDBL A-Z
2 DEFINT I, J, X

3 DIM B(9999), N(9999), A(9999), H(9999), L(9999), U(9999), X(9999), D(9999), P(9999), PS(9999), J(9999)

12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 6500
42 A(J44) = -5 + FIX(RND * 11)
43 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 6500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 6500)

183 REM R = (1 – RND * 2) * A(J)

187 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
189 REM X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP

251 SU = 0
254 FOR J44 = 1 TO 6499
258 SU = SU + X(J44)
266 NEXT J44

311 X(6500) = -X(1) – SU + (6500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 6500
355 PR = PR * X(J45)

359 NEXT J45

422 FOR J41 = 2 TO 6499
439 P(J41) = -ABS(X(J41) + SU + X(6500) – (6500 + 1))
427 NEXT J41
441 P(6500) = -ABS(PR – 1)

451 FOR J77 = 2 TO 6500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0

454 NEXT J77
577 SP = 0

578 FOR J99 = 2 TO 6500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 REM IF M < -99 THEN 1999
1947 PRINT A(1), A(2), A(3), A(4), A(6497), A(6498), A(6499), A(6500), 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=-31984 is shown below:

0 0 0 0 0
0 0 6500 -2 -32000

0 0 0 0 0
0 0 6503 -3 -31999

0 0 0 0 0
0 0 6502 -2 -31998

-1 -1 -1 -1 -1
-1 -1 13000 -2 -31007

1    1    1    1    1
1    1    1    0    -31996

2 2 2 2 2
2 2 -6497 -2 -31995

1 1 1 1 1
1 1 13000 -2 -31994

1 1 1 1 1
1 1 3 -5 -31993

1 1 1 1 1
1 1 12999 -3 -31992

-1 -1 -1 -1 -1
-1 -1 12999 -3 -31991

1 1 1 1 1
1 1 3 -3 -31990

1 1 1 1 1
1 1 3 -3 -31989

3 3 3 3 3
3 3 -12982 -17 -31988

1 1 1 1 1
1 1 13 -15 -31987

3 3 3 3 3
3 3 12991 -9 -31986

3 3 3 3 3
3 3 -12991 -8 -31985

1    1    1    1    1
1    1    1    0    -31984

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

Above at JJJJ=-31996 and at JJJJ= -31984, M=0. Of the 6500 unknowns, only the eight A’s of line 1947 are shown above.

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 for obtaining the output through JJJJ= -31984 was two hours and a half.

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] 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.

[3] 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.

[4] 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.

[5] 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.

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

[7] 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.

Solving in Integers a System of Simultaneous Nonlinear Diophantine Equations

Jsun Yui Wong

The following computer program seeks to solve in positive integers the following system of two simultaneous nonlinear Diophantine equations in three unknowns

a^3-b^3-c^3 = 3abc,

a^2 = 2(b+c).

This system comes from de Konnick and Mercier [1, p. 86].

One notes the starting vectors of line 52, which is 52 A(J44) = -100 + FIX(RND * 201).

0 REM DEFDBL A-Z
2 DEFINT I, J, X

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ

16 M = -1D+37
51 FOR J44 = 1 TO 3

52 A(J44) = -100 + FIX(RND * 201)
53 NEXT J44
128 FOR I = 1 TO 100

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 J = 1 + FIX(RND * 3)

183 R = (1 – RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP

212 REM

311 FOR J44 = 1 TO 3
313 IF X(J44) < 1 THEN 1670

319 NEXT J44

460 P1 = -ABS(X(1) ^ 3 – X(2) ^ 3 – X(3) ^ 3 – 3 * X(1) * X(2) * X(3))
461 P2 = -ABS(X(1) ^ 2 – 2 * (X(2) + X(3)))
462 IF P1 < 0 THEN P1 = P1 ELSE P1 = 0

463 IF P2 < 0 THEN P2 = P2 ELSE P2 = 0
466 P = P1 + P2

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 0 THEN 1999
1911 IF A(1) = 0 THEN 1999

1947 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31103 is shown below:

2 1 1 0 -31960
2 1 1 0 -31908
2 1 1 0 -31815
2 1 1 0 -31608
2 1 1 0 -31292
2 1 1 0 -31220
2 1 1 0 -31152
2 1 1 0 -31135
2 1 1 0 -31103

The solution on p. 295 of de Konnick and Mercier [1] is shown above.

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 [3], the wall-clock time for obtaining the output through JJJJ= -31103 was 2 seconds, not including “Creating .EXEC file…” time.

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. Americam Mathematical Society, Providence, Rhode Island, 2007.

[2] 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.

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

[4] 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.

Simultaneously Solving with Integers a System of Nonlinear Diophantine Equations

Jsun Yui Wong

The following computer program seeks to solve the following system of nonlinear Diophantine equations:

2 *x * (1 + y + y ^ 2) = 3 * (1 + y ^ 4)

2 * y * (1 + z + z ^ 2) = 3 * (1 + z ^ 4)

2 * z * (1 + x + x ^ 2) = 3 * (1 + x ^ 4).

This system comes from de Konnick and Mercier [1, p. 87].

One notes the starting vector of line 42, which is 42 A(J44) = -100 + FIX(RND * 201).

0 REM DEFDBL A-Z
2 DEFINT I, J, X

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 3
42 A(J44) = -100 + FIX(RND * 201)
43 NEXT J44
128 FOR I = 1 TO 100

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 3)

183 R = (1 – RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP

459 P1 = -ABS(2 * X(1) * (1 + X(2) + X(2) ^ 2) – 3 * (1 + X(2) ^ 4))

460 P2 = -ABS(2 * X(2) * (1 + X(3) + X(3) ^ 2) – 3 * (1 + X(3) ^ 4))

461 P3 = -ABS(2 * X(3) * (1 + X(1) + X(1) ^ 2) – 3 * (1 + X(1) ^ 4))
462 IF P1 < 0 THEN P1 = P1 ELSE P1 = 0

463 IF P2 < 0 THEN P2 = P2 ELSE P2 = 0

464 IF P3 < 0 THEN P3 = P3 ELSE P3 = 0

466 P = P1 + P2 + P3
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 0 THEN 1999
1947 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31913 is shown below:

1    1    1    0    -31988
1    1    1    0    -31945
1    1    1    0    -31917
1    1    1    0    -31913

The solution on p. 298 of de Konnick and Mercier [1] is shown above.

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 [3], the wall-clock time for obtaining the output through JJJJ= -31913 was 2 seconds, not including “Creating .EXEC file…” time..

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. Americam Mathematical Society, Providence, Rhode Island, 2007.

[2] 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.

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

[4] 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.