Substance

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--9th Powers and Twelve Unknowns

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation:

X(10)*X(1)^9+ X(2)^9
=X(3)^9+ X(4)^9+ X(11)* X(5)^9+ X(6)^9+ X(12)* X(7) ^9 + X(8)^9 + X(9)^9

This equation is based on Weisstein's 9.3.9, [15].

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 12
112 A(J44)=FIX( RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 12
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*12)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
172 IF (-X(10)*X(1)^9# + X(3)^9#+ X(4)^9#+ X(11)* X(5)^9#+ X(6)^9#+ X(12)* X(7) ^9# + X(8)^9# + X(9)^9# )<0# THEN 1670
190 PPPPP=(-X(10)*X(1)^9# + X(3)^9#+ X(4)^9#+ X(11)* X(5)^9#+ X(6)^9#+ X(12)* X(7) ^9# + X(8)^9# + X(9)^9# )^(1#/9#)
197 IF RND<.5 THEN X(2)=FIX(PPPPP) ELSE X(2)=FIX(PPPPP)+1#
212 FOR J44=1 TO 12
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>30 THEN X(J44)=10
215 NEXT J44
301 N(1)= X(10)*X(1)^9#+ X(2)^9#
304 N(2)=- X(3)^9#- X(4)^9#- X(11)* X(5)^9#- X(6)^9#- X(12)* X(7) ^9# - X(8)^9# - X(9)^9#
305 N(7)=N(1)+N(2)
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)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-1 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4)
1905 PRINT A(5),A(6),A(7),A(8)
1906 PRINT A(9),A(10),A(11),A(12)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31978 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

2 4 1 2
4 2 2 2
2 9 1 5
-1 -31998

1 4 1 1
4 1 1 1
1 12 1 7
0 -31988

1 11 1 1
1 1 1 11
1 12 1 7
0 -31980

2 6 2 6
2 2 2 2
2 10 1 5
0 -31978

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31978 was six seconds.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, "Diophantine Equation--9th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation9thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--10th Powers and Fourteen Unknowns

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation:

X(1)^10+ X(2)^10+ X(3)^10+ X(12)* X(4)^10+ X(5) ^10 + X(13)* X(6)^10 + X(7)^10 + X(8)^10+ X(9)^10
= X(10)^10+X(14)* X(11)^10

This equation is based on Weisstein's 10.3.24, [15].

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 14
112 A(J44)=FIX( RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 14
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*14)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
177 PPPPP=-X(1)^10#- X(2)^10#- X(3)^10#- X(12)* X(4)^10#-X(5) ^10# - X(13)* X(6)^10# - X(7)^10# - X(8)^10# +X(10)^10# + X(14)* X(11)^10#
179 IF (PPPPP)<0# THEN 1670
180 X(9)=(PPPPP )^(1#/10#)
212 FOR J44=1 TO 14
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>30 THEN X(J44)=10
215 NEXT J44
301 N(1)=X(1)^10#+ X(2)^10#+ X(3)^10#+ X(12)* X(4)^10#+ X(5) ^10# + X(13)* X(6)^10# + X(7)^10# + X(8)^10#+ X(9)^10#
303 N(2)=- X(10)^10#-X(14)* X(11)^10#
305 N(7)=N(1)+N(2)
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 14
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),A(4)
1905 PRINT A(5),A(6),A(7),A(8)
1906 PRINT A(9),A(10),A(11),A(12)
1907 PRINT A(13),A(14)
1908 REM
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31907 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

2 2 2 2
2 2 2 2
5 5 2 1
1 8
0 -31991

1 1 1 1
1 1 1 1
5 5 1 2
2 10
0 -31972

1 1 1 1
2 1 1 1
1 2 1 3
3 12
0 -31968

1 1 1 1
1 1 1 1
6 6 1 1
8 15
0 -31907

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31907 was eight seconds.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, "Diophantine Equation--10th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation10thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--10th Powers and Sixteen Unknowns, Another Instance

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 10.5.16 (54), Weisstein [15].

X(1)^10+ X(2)^10+ X(3)^10+ X(4)^10+ X(5) ^10
=X(13)* X(6)^10+X(7)^10+X(8)^10+ X(9)^10 +X(14)*X(10)^10 +X(15)*X(11)^10 + X(16)* X(12)^10

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 19
112 A(J44)=FIX( RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 19
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*16)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
177 PPPPP=( -X(1)^10# - X(2)^10#- X(3)^10#- X(4)^10#+X(13)* X(6)^10#+ X(7)^10# +X(8)^10#+X(9)^10# + X(14)* X(10)^10# +X(15)*X(11)^10# + X(16)*X(12)^10# )
179 IF (PPPPP)<0# THEN 1670
180 X(5)=(PPPPP )^(1#/10#)
212 FOR J44=1 TO 16
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>30 THEN X(J44)=10
215 NEXT J44
301 N(1)=X(1)^10#+ X(2)^10#+ X(3)^10#+ X(4)^10#+ X(5) ^10#
303 N(2)=-X(13)* X(6)^10#-X(7)^10#-X(8)^10#- X(9)^10# -X(14)*X(10)^10# -X(15)*X(11)^10# - X(16)* X(12)^10#
305 N(7)=N(1)+N(2)
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 19
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),A(4)
1905 PRINT A(5),A(6),A(7),A(8)
1906 PRINT A(9),A(10),A(11),A(12)
1907 PRINT A(13),A(14),A(15),A(16)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-27200 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

1 8 5 2
10 2 1 6
7 4 1 8
3 3 8 10
0 -30236

2 10 5 11
8 2 6 7
11 4 1 8
3 3 8 10
0 -28943

2 2 5 4
10 2 7 6
1 8 4 1
4 9 4 7
0 -27200

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-27200 was 12 minutes.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, "Diophantine Equation--10th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation10thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--10th Powers and Sixteen Unknowns

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 10.5.16 (53), Weisstein [15].

X(12)*X(1)^10+ X(2)^10+ X(13)* X(3)^10+ X(4)^10+ X(14)*X(5) ^10 + X(15)* X(6)^10 + X(7)^10
=X(16)* X(8)^10+X(9)^10+X(10)^10+ X(11)^10

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 19
112 A(J44)=FIX( RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 19
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*16)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
177 PPPPP=( -X(12)*X(1)^10# - X(2)^10#-X(13)* X(3)^10#- X(4)^10#-X(14)* X(5)^10#-X(15)* X(6)^10#+ X(16)* X(8)^10# +X(9)^10#+X(10)^10#+X(11)^10#)
179 IF (PPPPP)<0# THEN 1670
180 X(7)=(PPPPP )^(1#/10#)
212 FOR J44=1 TO 16
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>30 THEN X(J44)=10
215 NEXT J44
301 N(1)=X(12)*X(1)^10#+ X(2)^10#+ X(13)* X(3)^10#+ X(4)^10#+ X(14)*X(5) ^10# + X(15)* X(6)^10# + X(7)^10#
303 N(2)=-X(16)* X(8)^10#-X(9)^10#-X(10)^10#- X(11)^10#
305 N(7)=N(1)+N(2)
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 19
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),A(4)
1905 PRINT A(5),A(6),A(7),A(8)
1906 PRINT A(9),A(10),A(11),A(12)
1907 PRINT A(13),A(14),A(15),A(16)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31965 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

3 1 3 1
1 1 4 1
4 3 3 1
1 10 5 17
0 -31990

1 2 2 2
2 2 3 2
2 3 2 1
1 1 1 3
-1 -31981

2 3 1 2
2 2 4 2
1 4 3 2
1 5 3 11
0 -31971

3 1 3 1
3 1 6 3
1 1 6 4
3 3 1 10
-1 -31965

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31965 was six seconds.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, "Diophantine Equation--10th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation10thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--9th Powers and Twenty-Three Unknowns, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 9.11.12 [15].

X(1)^9+X(2)^9+X(3)^9+X(4)^9+X(5) ^9 +X(6)^9 +X(7)^9 + X(8) ^9 +X(9)^9 +X(10)^9+X(11)^9
=X(20)^9+X(21)^9+X(12)^9+X(13)^9+X(14) ^9 +X(15)^9 +X(16)^9 + X(17) ^9 +X(18)^9 +X(19)^9 +X(22)^9+X(23)^9

While line 214 of the earlier edition is 214 IF X(J44)>75 THEN X(J44)=30, line 214 here is
214 IF X(J44)>100 THEN X(J44)=30.

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 23
111 REM
112 A(J44)=FIX( RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 23
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*7)
140 B=1+FIX(RND*23)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
177 PPPPA=(-X(2)^9#-X(3)^9#-X(4)^9#-X(5)^9#-X(6)^9#-X(7)^9#-X(8)^9#-X(9)^9#-X(10)^9#-X(11)^9#)
178 PPPPB=(X(20)^9#+X(21)^9#+X(22)^9#+X(23)^9#+X(12)^9#+X(13)^9#+X(14)^9#+X(15)^9#+X(16)^9#+X(17)^9#+X(18)^9#+X(19)^9#)
179 IF (PPPPA+PPPPB)<0# THEN 1670
180 PPPPP=(PPPPA+PPPPB )^(1#/9#)
181 IF RND<.5 THEN X(1)=FIX(PPPPP) ELSE X(1)=FIX(PPPPP)+1
212 FOR J44=1 TO 23
213 IF X(J44)<1 THEN X(J44)=30
214 IF X(J44)>100 THEN X(J44)=30
215 NEXT J44
301 N(1)=X(1)^9#+X(2)^9#+X(3)^9#+X(4)^9#+X(5) ^9# +X(6)^9# +X(7)^9# + X(8) ^9# +X(9)^9# +X(10)^9#+X(11)^9#
303 N(2)=-X(20)^9#-X(21)^9#-X(12)^9#-X(13)^9#-X(14) ^9# -X(15)^9# -X(16)^9# - X(17) ^9# -X(18)^9# -X(19)^9# -X(22)^9#-X(23)^9#
305 N(7)=N(1)+N(2)
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 23
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1890 FOR J44=1 TO 23
1891 IF A(J44)>74 THEN 1999
1892 NEXT J44
1904 PRINT A(1),A(2),A(3),A(4)
1905 PRINT A(5),A(6),A(7),A(8)
1906 PRINT A(9),A(10),A(11),A(12)
1907 PRINT A(13),A(14),A(15),A(16)
1908 PRINT A(17),A(18),A(19),A(20)
1909 PRINT A(21),A(22),A(23)
1910 REM
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=613 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

36 30 4 5
30 6 4 36
30 6 5 10
12 3 8 12
3 8 3 40
9 11 13
0 613

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=613 was three hours and fifty minutes.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, Diophantine Equation--9th Powers. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation9thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--6th Powers and Fifteen Variables

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 6.7.8 [15].

X(1)^6+X(2)^6+X(3)^6+X(4)^6+X(5) ^6 +X(6)^6 +X(7)^6
= X(8)^6 +X(9)^6 +X(10)^6+X(11)^6+X(12)^6+X(13)^6+X(14) ^6 +X(15)^6

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 15
112 A(J44)=FIX( RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 15
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*15)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
171 IF (-X(2)^6#-X(3)^6#-X(4)^6#-X(5)^6#-X(6)^6#-X(7)^6#+X(8)^6#+X(9)^6#+X(10)^6#+X(11)^6#+X(12)^6#+X(13)^6#+X(14)^6#+X(15)^6#)<0# THEN 1670
175 PPPPP=(-X(2)^6#-X(3)^6#-X(4)^6#-X(5)^6#-X(6)^6#-X(7)^6#+X(8)^6#+X(9)^6#+X(10)^6#+X(11)^6#+X(12)^6#+X(13)^6#+X(14)^6#+X(15)^6#)^(1#/6#)
181 IF RND<.5 THEN X(1)=FIX(PPPPP) ELSE X(1)=FIX(PPPPP)+1
212 FOR J44=1 TO 15
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>50 THEN X(J44)=10
215 NEXT J44
301 N(1)=X(1)^6#+X(2)^6#+X(3)^6#+X(4)^6#+X(5) ^6# +X(6)^6# +X(7)^6#
303 N(2)=-X(10)^6#-X(11)^6#-X(12)^6#-X(13)^6#-X(14) ^6# -X(15)^6# -X(8)^6# -X(9)^6#
305 N(7)=N(1)+N(2)
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 15
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)
1906 PRINT A(9),A(10),A(11),A(12)
1907 PRINT A(13),A(14),A(15)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-29550 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

8 5 2 11
6 5 5 3
1 4 10 4
3 10 7
0 -31918

13 2 1 4
9 1 4 1
12 1 3 8
10 10 7
0 -30276

4 3 1 1
1 10 1 8
5 7 5 6
2 9 5
0 -29586

10 10 8 13
3 1 10 4
2 14 4 4
5 3 9
0 -29550

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-29550 was five minutes and ten seconds.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, Diophantine Equation--6th Powers. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation6thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--8th Powers and Nineteen Variables, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 8.9.10 [15].

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

While line 112 of the preceding paper is 112 A(J44)=FIX( RND *10 ), line 112 here is
112 A(J44)=FIX( RND *5 ).

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 19
112 A(J44)=FIX( RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 19
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*19)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
171 IF (-X(2)^7#-X(3)^7#-X(4)^7#-X(5)^7#-X(6)^7#-X(7)^7#-X(8)^7#-X(9)^7#+X(10)^7#+X(11)^7#+X(12)^7#+X(13)^7#+X(14)^7#+X(15)^7#+X(16)^7#+X(17)^7#+X(18)^7#+X(19)^7#)<0# THEN 1670
175 PPPPP=(-X(2)^7#-X(3)^7#-X(4)^7#-X(5)^7#-X(6)^7#-X(7)^7#-X(8)^7#-X(9)^7#+X(10)^7#+X(11)^7#+X(12)^7#+X(13)^7#+X(14)^7#+X(15)^7#+X(16)^7#+X(17)^7#+X(18)^7#+X(19)^7#)^(1#/8#)
181 IF RND<.5 THEN X(1)=FIX(PPPPP) ELSE X(1)=FIX(PPPPP)+1
212 FOR J44=1 TO 19
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>50 THEN X(J44)=10
215 NEXT J44
301 N(1)=X(1)^8#+X(2)^8#+X(3)^8#+X(4)^8#+X(5) ^8# +X(6)^8# +X(7)^8# + X(8) ^8# +X(9)^8#
303 N(2)=-X(10)^8#-X(11)^8#-X(12)^8#-X(13)^8#-X(14) ^8# -X(15)^8# -X(16)^8# - X(17) ^8# -X(18)^8# -X(19)^8#
305 N(7)=N(1)+N(2)
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 19
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)
1906 PRINT A(9),A(10),A(11),A(12)
1907 PRINT A(13),A(14),A(15),A(16)
1908 PRINT A(17),A(18),A(19)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31795 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

6 2 3 10
5 3 3 5
5 9 2 1
1 9 8 1
2 1 2
0 -31924

5 6 10 5
4 3 3 3
5 1 2 9
4 1 1 8
1 9 2
0 -31874

5 2 5 6
3 3 3 5
10 2 1 2
2 1 1 9
1 9 8
0 -31850

5 3 5 6
10 3 3 5
2 1 9 9
2 2 2 1
1 8 1
0 -31795

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31795 was forty seconds.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, Diophantine Equation--8th Powers. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation8thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--8th Powers and Nineteen Variables

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 8.9.10 [15].

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

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 19
112 A(J44)=FIX( RND *10 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 19
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*19)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
171 IF (-X(2)^7#-X(3)^7#-X(4)^7#-X(5)^7#-X(6)^7#-X(7)^7#-X(8)^7#-X(9)^7#+X(10)^7#+X(11)^7#+X(12)^7#+X(13)^7#+X(14)^7#+X(15)^7#+X(16)^7#+X(17)^7#+X(18)^7#+X(19)^7#)<0# THEN 1670
175 PPPPP=(-X(2)^7#-X(3)^7#-X(4)^7#-X(5)^7#-X(6)^7#-X(7)^7#-X(8)^7#-X(9)^7#+X(10)^7#+X(11)^7#+X(12)^7#+X(13)^7#+X(14)^7#+X(15)^7#+X(16)^7#+X(17)^7#+X(18)^7#+X(19)^7#)^(1#/8#)
181 IF RND<.5 THEN X(1)=FIX(PPPPP) ELSE X(1)=FIX(PPPPP)+1
212 FOR J44=1 TO 19
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>50 THEN X(J44)=10
215 NEXT J44
301 N(1)=X(1)^8#+X(2)^8#+X(3)^8#+X(4)^8#+X(5) ^8# +X(6)^8# +X(7)^8# + X(8) ^8# +X(9)^8#
303 N(2)=-X(10)^8#-X(11)^8#-X(12)^8#-X(13)^8#-X(14) ^8# -X(15)^8# -X(16)^8# - X(17) ^8# -X(18)^8# -X(19)^8#
305 N(7)=N(1)+N(2)
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 19
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)
1906 PRINT A(9),A(10),A(11),A(12)
1907 PRINT A(13),A(14),A(15),A(16)
1908 PRINT A(17),A(18),A(19)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-30446 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

5 6 5 10
3 3 10 5
3 1 1 10
9 9 2 2
1 1 8
0 -31034

6 10 3 3
5 5 7 3
5 2 9 1
8 1 1 2
1 9 7
0 -30911

5 3 6 5
3 10 3 10
5 9 2 9
1 1 8 10
1 2 1
0 -30619

5 3 3 5
9 10 5 3
6 1 1 1
2 2 9 1
9 8 9
0 -30446

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-30446 was six minutes.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Eric W. Weisstein, Diophantine Equation--8th Powers. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation8thPowers.html

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--7th Powers and Twelve Variables, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 7.6.6 [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 +X(9)^7 +X(10)^7 +X(11) ^7+X(12)^7

While line 214 of the preceding paper is 214 IF X(J44)>30 THEN X(J44)=10, here line 214 is
214 IF X(J44)>35 THEN X(J44)=10.

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 16
112 A(J44)=FIX( RND *20 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 16
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*12)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
169 IF (-X(1)^7#-X(2)^7#-X(3)^7#-X(4)^7#-X(5) ^7# +X(7)^7# + X(8) ^7# +X(9)^7# +X(10)^7# +X(11) ^7#+X(12)^7# )<0# THEN 1670
173 PPPPP=(-X(1)^7#-X(2)^7#-X(3)^7#-X(4)^7#-X(5) ^7# +X(7)^7# + X(8) ^7# +X(9)^7# +X(10)^7# +X(11) ^7#+X(12)^7# )^(1#/7#)
181 IF RND<.5 THEN X(6)=FIX(PPPPP) ELSE X(6)=FIX(PPPPP)+1
212 FOR J44=1 TO 12
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>35 THEN X(J44)=10
215 NEXT J44
219 IF (X(1)^2#+X(2)^2#+X(3)^2#+X(4)^2#+X(5)^2#+X(6)^2# -X(7)^2# - X(8)^2#-X(9)^2# -X(10)^2# -X(11)^2# -X(12)^2#) =0# THEN 1670
222 N(7)=X(1)^7#+X(2)^7#+X(3)^7#+X(4)^7#+X(5) ^7# +X(6)^7# -X(7)^7# - X(8) ^7# -X(9)^7# -X(10)^7# -X(11) ^7#-X(12)^7#
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 16
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)
1906 PRINT A(9),A(10),A(11),A(12)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through through JJJJ=8716 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

5 9 15 20
1 13 10 18
6 8 19 2
0 -26434

3 13 10 2
6 6 1 7
12 7 12 1
0 -26050

13 19 10 8
8 16 15 12
18 17 10 2
0 -4758

15 7 2 12
17 18 13 19
8 16 8 7
0 765

8 13 19 4
8 16 15 4
18 12 2 17
0 8640

3 22 9 7
7 20 3 14
4 16 8 23
0 8716

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=8716 was three hours.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] 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 Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html

Testing the Nonlinear Integer/Continuous Programming Solver with a Diophantine Equation--7th Powers and Twelve Variables, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation, which is based on Weisstein's 7.6.6 [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 +X(9)^7 +X(10)^7 +X(11) ^7+X(12)^7

While line 214 of the preceding paper is 214 IF X(J44)>23 THEN X(J44)=10, here line 214 is
214 IF X(J44)>30 THEN X(J44)=10.

0 DEFDBL A-Z
1 DEFINT I,J,K,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
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 16
112 A(J44)=FIX( RND *20 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 16
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*12)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 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
169 IF (-X(1)^7#-X(2)^7#-X(3)^7#-X(4)^7#-X(5) ^7# +X(7)^7# + X(8) ^7# +X(9)^7# +X(10)^7# +X(11) ^7#+X(12)^7# )<0# THEN 1670
173 PPPPP=(-X(1)^7#-X(2)^7#-X(3)^7#-X(4)^7#-X(5) ^7# +X(7)^7# + X(8) ^7# +X(9)^7# +X(10)^7# +X(11) ^7#+X(12)^7# )^(1#/7#)
181 IF RND<.5 THEN X(6)=FIX(PPPPP) ELSE X(6)=FIX(PPPPP)+1
212 FOR J44=1 TO 12
213 IF X(J44)<1 THEN X(J44)=10
214 IF X(J44)>30 THEN X(J44)=10
215 NEXT J44
219 IF (X(1)^2#+X(2)^2#+X(3)^2#+X(4)^2#+X(5)^2#+X(6)^2# -X(7)^2# - X(8)^2#-X(9)^2# -X(10)^2# -X(11)^2# -X(12)^2#) =0# THEN 1670
222 N(7)=X(1)^7#+X(2)^7#+X(3)^7#+X(4)^7#+X(5) ^7# +X(6)^7# -X(7)^7# - X(8) ^7# -X(9)^7# -X(10)^7# -X(11) ^7#-X(12)^7#
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 16
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-2 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4)
1905 PRINT A(5),A(6),A(7),A(8)
1906 PRINT A(9),A(10),A(11),A(12)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through through JJJJ=-22627 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

5 9 15 20
1 13 10 18
6 8 19 2
0 -26434

8 9 4 11
8 16 13 1
15 6 13 6
-2 -23309

16 8 8 5
19 13 12 15
17 2 18 5
0 -22627

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through
JJJJ=-22627 was 32 minutes.

Acknowledgement

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 (1979), Mathematical Circus. Knopf (1979).

[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). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the 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] 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 Brick. en.wikipedia.org/wiki/Euler_brick

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html