Substance

A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2, Second Edition

Jsun Yui Wong

Based on the computer program of the original edition [9], the computer program below seeks to solve simultaneously the system of three Diophantine equations taken from page 112 of Sierpinski [6]. His simultaneous equations are as follows:

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

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 6
112 A(J44)=50+ ( RND *1000)
113 NEXT J44
128 FOR I=1 TO 2000
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*6)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
164 REM IF RND<.5 THEN X(B)=(A(B)-RND*10) ELSE X(B)=(A(B) +RND*10)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
185 FOR J44=1 TO 6
186 IF X(J44)<1 THEN 1670
187 NEXT J44
191 X(4)= ( +X(1)^2 +X(2)^2 ) ^(1#/2#)
192 X(5)= ( +X(1)^2 +X(3)^2 ) ^(1#/2#)
193 X(6)= ( +X(2)^2 +X(3)^2 ) ^(1#/2#)
194 N(7)= X(4)^2-X(1)^2 -X(2)^2
195 N(8)= X(5)^2-X(1)^2 -X(3)^2
196 N(9)= X(6)^2-X(2)^2 -X(3)^2
197 GOTO 322
201 FOR J44=7 TO 9
202 IF ABS(N(J44))>0 THEN PN(J44)=-1000000!*ABS(N(J44)) ELSE PN(J44)=0
203 NEXT J44
321 REM
322 PD1=-ABS(N(7))-ABS(N(8) )-ABS( N(9) )
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 6
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),A(5),A(6),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=-26378 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

160 792 231 808 281
825 0 -31975

132 720 85 732 157
725 0 -29865

480 88 234 488 534
250 0 -29583

234 88 480 250 534
488 0 -29034

275 252 240 373 365
348 0 -28410

264 1440 170 1464 314
1450 0 -27202

117  240  44  267  125
244  0  -26378

The last solution shown above (117  240  44  267  125  244), the smallest Euler brick, was discovered by Paul Halcke in 1719, Wikipedia [8].

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=-26378 was 30 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] M. Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

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

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

[8] Wikipedia, Euler Brick.  http://en.wikipedia.org/wiki/Euler_brick.

[9] 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.

The Domino Method Applied to Solving a System of Twenty General Integer Variables and Ten Simultaneous Nonlinear Equations, Including Five Exponential Diophantine Equations

Jsun Yui Wong

The following computer program seeks to solve simultaneously the following system of ten simultaneous nonlinear equations, including five exponential Diophantine equations:

1+ (2*3 )^X(1) = 2^X(3)+3^X(4)
X(5)^X(1)+X(6)^ X(2) =X(7)^X(3)+X(8)^X(4)
X(9)^X(1)+X(10)^ X(4) =X(11)^X(3)+X(12)^X(2)
1 + X(13)^X(1)+X(14)^ X(4) =X(15)^X(3)+X(16)^X(2)
X(17)^X(1)+X(18)^ X(2) +X(19)^X(4) =X(20)^X(3)
X(1)^2+X(2)^2 +X(3)^2 +X(4)^2 = 6
X(5)^2+X(6)^2 +X(7)^2 +X(8)^2 =252
X(9)^2+X(10)^2 +X(11)^2 +X(12)^2 =108
X(13)^2+X(14)^2 +X(15)^2 +X(16)^2 =87
X(17)^2+X(18)^2 +X(19)^2 +X(20)^2 =68

The first exponential Diophantine equation shown above is 1+ (2*3)^X(1) = 2^X(3)+3^X(4), which is based on 1+ (pq)^a = p^b +q^c on page 298 of Brenner and Foster [2], where p and q are given distinct prime numbers; "... does not seem to be amenable to our methods," Brenner and Foster [2, page 298]. The other four exponential Diophantine equations are X(5)^X(1)+X(6)^ X(2) =X(7)^X(3)+X(8)^X(4), X(9)^X(1)+X(10)^ X(4) =X(11)^X(3)+X(12)^X(2), 1 +X(13)^X(1)+X(14)^ X(4) =X(15)^X(3)+X(16)^X(2), and X(17)^X(1)+X(18)^ X(2) +X(19)^X(4)=X(20)^X(3), which are based on
13^X(1)+7^ X(2) =3^X(3)+5^X(4), 5^X(1)+5^ X(2)=3^X(3)+7^X(4),
1 +2^X(1)+7^ X(2) =3(15)^X(3)+5^X(4), and 5^X(1)+5^ X(2) +3^X(3)=3^X(4) on page 744 of Perez, Amaya, and Correa [5], respectively.

Line 1, line 1455, and line 1456 are noteworthy.

0 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)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
6 DIM Y(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 20
112 A(J44)=RND *5
114 NEXT J44
128 FOR I=1 TO 1500
129 FOR KKQQ=1 TO 20
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*20)
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
198 IF ( 6#-X(2)^2 -X(3)^2 -X(4)^2 ) <0 THEN 1670
201 X(1)= ( 6#-X(2)^2 -X(3)^2 -X(4)^2 )^.5#
208 IF ( 252#-X(6)^2 -X(7)^2 -X(8)^2 ) <0 THEN 1670
211 X(5)= ( 252#-X(6)^2 -X(7)^2 -X(8)^2 )^.5#
213 IF ( 108#-X(10)^2 -X(11)^2 -X(12)^2 ) <0 THEN 1670
215 X(9)= ( 108#-X(10)^2 -X(11)^2 -X(12)^2 )^.5#
230 IF ( 87#-X(14)^2 -X(15)^2 -X(16)^2 ) <0 THEN 1670
233 X(13)= ( 87#-X(14)^2 -X(15)^2 -X(16)^2 )^.5#
240 IF ( 68#-X(18)^2 -X(19)^2 -X(20)^2 ) <0 THEN 1670
244 X(17)= ( 68#-X(18)^2 -X(19)^2 -X(20)^2 )^.5#
369 FOR J44=1 TO 20
370 IF X(J44)<0 THEN X(J44)=15
372 NEXT J44
382 N(1)= 1#+ (2#*3# )^X(1) -2#^X(3)-3#^X(4)
388 N(2)= X(5)^X(1)+X(6)^ X(2) -X(7)^X(3)-X(8)^X(4)
393 N(3)= X(9)^X(1)+X(10)^ X(4) -X(11)^X(3)-X(12)^X(2)
397 N(4)= 1 + X(13)^X(1)+X(14)^ X(4) -X(15)^X(3)-X(16)^X(2)
398 N(5)= X(17)^X(1)+X(18)^ X(2) +X(19)^X(4)-X(20)^X(3)
922 PD1A=-ABS(N(1)) -ABS(N(2)) -ABS(N(3)) -ABS(N(4)) -ABS(N(5))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 20
1455 A(KLX)=CINT(X(KLX) )
1456 Y(KLX)=X(KLX)
1459 NEXT KLX
1551 FT= + 1#+ ( 2#*3# )^A(1)
1552 ST=-2#^A(3) -3#^A(4)
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1895 PRINT A(1),A(2),A(3),A(4)
1897 PRINT Y(1),Y(2),Y(3),Y(4)
1903 PRINT A(5),A(6),A(7),A(8)
1904 PRINT Y(5),Y(6),Y(7),Y(8)
1905 PRINT A(9),A(10),A(11),A(12)
1906 PRINT Y(9),Y(10),Y(11),Y(12)
1908 PRINT A(13),A(14),A(15),A(16)
1909 PRINT Y(13),Y(14),Y(15),Y(16)
1911 PRINT A(17),A(18),A(19),A(20)
1912 PRINT Y(17),Y(18),Y(19),Y(20)
1918 PRINT M,JJJJ
1919 PRINT FT,ST
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=-8831 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

1 0 2 1
1 0 2 1
13 7 3 5
13 7 3 5
5 5 3 7
5 5 3 7
7 2 3 5
7 2 3 5
7 3 1 3
7 3 1 3
0 -22426
7 -7

1 0 2 1
1 0 2 1
13 7 3 5
13 7 3 5
5 5 3 7
5 5 3 7
7 2 3 5
7 2 3 5
7 3 1 3
7 3 1 3
0 -21737
7 -7

1 0 2 1
1 0 2 1
13 7 3 5
13 7 3 5
5 5 3 7
5 5 3 7
7 2 3 5
7 2 3 5
5 5 3 3
5 5 3 3
0 -8831
7 -7

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=-8831 was one hour and a half.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), Pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

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

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

Finding Integer Solutions of Nonlinear Systems of Simultaneous Equations with the Domino Method, Part 3

Jsun Yui Wong

The following computer program seeks to solve the following nonlinear system of six equations, including the exponential Diophantine equation X(5)^X(1) +X(6)^X(2)
= X(7)^X(3)+X(8)^X(4) + 2, which is based on 3^X(1) + 7^X(2) = 3^X(3)+ 5^X(4) + 2
on page 285 of Brenner and Foster [2].

X(5) + X(6)+X(7) +X(8) )= 18
X(1)^2 +X(2)^2 + X(3)^2 +X(4)^2 = 78
X(5)^2 +X(6)^2 + X(7)^2 +X(8)^2 = 92
X(5)^X(1) +X(6)^X(2) = X(7)^X(3)+X(8)^X(4) + 2
X(1)^3 +X(2)^3 + X(3)^3 +X(4)^3 = 406
X(5)^3 +X(6)^3 + X(7)^3 +X(8)^3 = 522

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 8
112 A(J44)=FIX(RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 8
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8)
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
211 X(5)= 18# - X(7)-X(8) -X(6)
369 FOR J44=1 TO 8
370 IF X(J44)<0 THEN X(J44)=10
372 NEXT J44
381 N(1)= - X(7)^X(3)-X(8)^X(4) -2# + X(5)^X(1) +X(6)^X(2)
383 N(2)= -78# + X(1)^2# +X(2)^2# + X(3)^2# +X(4)^2#
384 N(3)=-92# + X(5)^2# +X(6)^2# + X(7)^2# +X(8)^2#
386 N(4)=-406# + X(1)^3# +X(2)^3# + X(3)^3# +X(4)^3#
388 N(5)=-522# + X(5)^3# +X(6)^3# + X(7)^3# +X(8)^3#
922 PD1A=-ABS(N(1)) -ABS(N(2)) -ABS( N(3)) -ABS(N(4)) -ABS(N(5))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 PRINT A(1),A(2),A(3),A(4)
1904 PRINT A(5),A(6),A(7),A(8)
1907 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=-27503 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

4 6 5 1
7 3 5 3
0 -28227

4 6 5 1
7 3 5 3
0 -27616

4 6 5 1
7 3 5 3
0 -27503

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=-27503 was eight 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] M. Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

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

Finding Integer Solutions of Nonlinear Systems of Simultaneous Equations with the Domino Method, Part 2

Jsun Yui Wong

The following computer program seeks to solve the following nonlinear system of five equations, including the exponential Diophantine equation X(5)^X(1) +X(6)^X(2)
= X(7)^X(3)+X(8)^X(4) + 2, which is based on 3^X(1) + 7^X(2) = 3^X(3)+ 5^X(4) + 2
on page 285 of Brenner and Foster [2].

X(5) + X(6)+X(7) +X(8) )= 18
X(1)^2 +X(2)^2 + X(3)^2 +X(4)^2 = 78
X(5)^2 +X(6)^2 + X(7)^2 +X(8)^2 = 92
X(5)^X(1) +X(6)^X(2) = X(7)^X(3)+X(8)^X(4) + 2
X(1)^3 +X(2)^3 + X(3)^3 +X(4)^3 = 406

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 8
112 A(J44)=FIX(RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 8
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8)
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
211 X(5)= 18# - X(7)-X(8) -X(6)
369 FOR J44=1 TO 8
370 IF X(J44)<0 THEN X(J44)=10
372 NEXT J44
381 N(1)= - X(7)^X(3)-X(8)^X(4) -2# + X(5)^X(1) +X(6)^X(2)
383 N(2)= -78# + X(1)^2# +X(2)^2# + X(3)^2# +X(4)^2#
384 N(3)=-92# + X(5)^2# +X(6)^2# + X(7)^2# +X(8)^2#
386 N(4)=-406# + X(1)^3# +X(2)^3# + X(3)^3# +X(4)^3#
922 PD1A=-ABS(N(1)) -ABS(N(2)) -ABS( N(3)) -ABS(N(4))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 PRINT A(1),A(2),A(3),A(4)
1904 PRINT A(5),A(6),A(7),A(8)
1907 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=-27652 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

4 6 5 1
7 3 5 3
0 -30093

4 6 5 1
7 3 5 3
0 -28268

4 6 5 1
7 3 5 3
0 -27652

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=-27652 was seven 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] M. Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

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

Finding Integer Solutions of Nonlinear Systems of Simultaneous Equations with the Domino Method

Jsun Yui Wong

The following computer program seeks to solve the following nonlinear system of equations, including the exponential Diophantine equation X(5)^X(1) +X(6)^X(2)
= X(7)^X(3)+X(8)^X(4) + 2, which is based on 3^X(1) + 7^X(2) = 3^X(3)+ 5^X(4) + 2
on page 285 of Brenner and Foster [2].

X(5) + X(6)+X(7) +X(8) )= 18
X(1)^2 +X(2)^2 + X(3)^2 +X(4)^2 = 78
X(5)^2 +X(6)^2 + X(7)^2 +X(8)^2 = 92
X(5)^X(1) +X(6)^X(2) = X(7)^X(3)+X(8)^X(4) + 2

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 8
112 A(J44)=FIX(RND *5 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 8
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8)
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
211 X(5)= 18# - X(7)-X(8) -X(6)
369 FOR J44=1 TO 8
370 IF X(J44)<0 THEN X(J44)=10
372 NEXT J44
381 N(1)= - X(7)^X(3)-X(8)^X(4) -2# + X(5)^X(1) +X(6)^X(2)
383 N(2)= -78# + X(1)^2# +X(2)^2# + X(3)^2# +X(4)^2#
384 N(3)=-92# + X(5)^2# +X(6)^2# + X(7)^2# +X(8)^2#
922 PD1A=-ABS(N(1)) -ABS(N(2)) -ABS( N(3))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 8
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 PRINT A(1),A(2),A(3),A(4)
1904 PRINT A(5),A(6),A(7),A(8)
1907 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=-14348 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

4 6 1 5
7 3 3 5
0 -29853

4 6 5 1
7 3 5 3
0 -14645

4 6 1 5
7 3 3 5
0 -14348

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=-14348 was fifteen 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] M. Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

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

A Computer Program for Solving an Instance of 1 + ( pq )^a = p^b + q^c, Part 5

Jsun Yui Wong

The following computer program seeks to solve 1+ ( 5*31 )^x(1) = 5^x(2) + 31^x(3), which is based on the exponential Diophantine equation 1 + ( pq )^a = p^b + q^c on page 298 of Brenner and Foster [2], where p and q are given distinct primes, [2].

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 3
112 A(J44)=-5+FIX(RND *10 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*3)
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
369 FOR J44=1 TO 3
370 IF X(J44)<-10 THEN X(J44)=-10
371 IF X(J44)>10 THEN X(J44)=10
372 NEXT J44
381 N(1)=-5#^X(2) -31#^X(3) + 1#+ (5#*31# )^X(1)
922 PD1A=-ABS(N(1))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1551 FT= + 1#+ ( 5#*31# )^A(1)
1552 ST=-5#^A(2) -31#^A(3)
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 PRINT A(1),A(2),A(3),M,JJJJ
1903 PRINT FT,ST
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=-31955 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

0 0 0 0 -31997
2 -2

1 3 1 0 -31992
156 -156

1 3 1 0 -31973
156 -156

1 3 1 0 -31955
156 -156

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=-31955 was two 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.

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

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

A Computer Program for Solving an Instance of 1 + ( pq )^a = p^b + q^c, Part 4

Jsun Yui Wong

The following computer program seeks to solve 1+ ( 2*31 )^x(1) = 2^x(2) + 31^x(3), which is based on the exponential Diophantine equation 1 + ( pq )^a = p^b + q^c on page 298 of Brenner and Foster [2], where p and q are given distinct primes, [2].

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 3
112 A(J44)=-5+FIX(RND *10 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*3)
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
369 FOR J44=1 TO 3
370 IF X(J44)<-10 THEN X(J44)=-10
371 IF X(J44)>10 THEN X(J44)=10
372 NEXT J44
381 N(1)=-2#^X(2) -31#^X(3) + 1#+ (2#*31# )^X(1)
922 PD1A=-ABS(N(1))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1551 FT= + 1#+ ( 2#*31# )^A(1)
1552 ST=-2#^A(2) -31#^A(3)
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 PRINT A(1),A(2),A(3),M,JJJJ
1903 PRINT FT,ST
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=-31938 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

0 0 0 0 -31991
2 -2

0 0 0 0 -31975
2 -2

0 0 0 0 -31973
2 -2

1 5 1 0 -31967
63 -63

0 0 0 0 -31963
2 -2

0 0 0 0 -31961
2 -2

1 5 1 0 -31938
63 -63

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=-31938 was three 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.

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

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

A Computer Program for Solving an Instance of 1 + ( pq )^a = p^b + q^c, Part 3

Jsun Yui Wong

The following computer program seeks to solve 1+ ( 2*7 )^x(1) = 2^x(2) + 7^x(3), which is based on the exponential Diophantine equation 1 + ( pq )^a = p^b + q^c on page 298 of Brenner and Foster [2], where p and q are given distinct primes, [2].

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 3
112 A(J44)=-5+FIX(RND *10 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*3)
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
369 FOR J44=1 TO 3
370 IF X(J44)<-10 THEN X(J44)=-10
371 IF X(J44)>10 THEN X(J44)=10
372 NEXT J44
381 N(1)=-2#^X(2) -7#^X(3) + 1#+ (2#*7# )^X(1)
922 PD1A=-ABS(N(1))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1551 FT= + 1#+ ( 2#*7# )^A(1)
1552 ST=-2#^A(2) -7#^A(3)
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 PRINT A(1),A(2),A(3),M,JJJJ
1903 PRINT FT,ST
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=-31964 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

0 0 0 0 -31997
2 -2

1 3 1 0 -31995
15 -15

1 3 1 0 -31991
15 -15

0 0 0 0 -31985
2 -2

1 3 1 0 -31983
15 -15

1 3 1 0 -31964
15 -15

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=-31964 was two 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] M. Gardner (1979), Mathematical Games.  Scientific American, 241 (3), Page 25.

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

A Computer Program for Solving an Instance of 1 + ( pq )^a = p^b + q^c, Part 2

Jsun Yui Wong

The following computer program seeks to solve 1+ ( 3*13 )^x(1) = 3^x(2) + 13 3^x(3), which is based on the exponential Diophantine equation 1 + ( pq )^a = p^b + q^c on page 298 of Brenner and Foster [2], where p and q are given distinct primes, [2].

0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 3
112 A(J44)=-5+FIX(RND *10 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*3)
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
369 FOR J44=1 TO 3
370 IF X(J44)<-10 THEN X(J44)=-10
371 IF X(J44)>10 THEN X(J44)=10
372 NEXT J44
381 N(1)=-3#^X(2) -13#^X(3) + 1#+ (3#*13# )^X(1)
922 PD1A=-ABS(N(1))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1551 FT= + 1#+ ( 3#*13 )^A(1)
1552 ST=-3#^A(2) -13#^A(3)
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 PRINT A(1),A(2),A(3),M,JJJJ
1903 PRINT FT,ST
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=-31966 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

0 0 0 0 -31996
2 -2

1 3 1 0 -31986
40 -40

0 0 0 0 -31981
2 -2

0 0 0 0 -31974
2 -2

0 0 0 0 -31972
2 -2

1 3 1 0 -31971
40 -40

1 3 1 0 -31970
40 -40

1 3 1 0 -31966
40 -40

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=-31966 was two 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] 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.

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

A Computer Program for Solving x(1)^x(2) = x(2)^x(1)

Jsun Yui Wong

The following computer program seeks to solve  x(1)^x(2) = x(2)^x(1), which is based on the exponential Diophantine equation x^y=y^x on page 299 of Brenner and Foster [2] and on page 25 of Gardner [3].

0 DEFDBL A-Z
1 DEFINT I,J,K,X,A
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 7
112 A(J44)=-5+FIX(RND *10 )
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*7)
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
369 FOR J44=1 TO 7
370 IF X(J44)<-10 THEN X(J44)=10
371 IF X(J44)>10 THEN X(J44)=10
372 NEXT J44
381 N(1)=-X(2) ^(X(1) ) + X(1)^(X(2) )
922 PD1A=-ABS(N(1))
929 PD1=PD1A
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1902 LPRINT A(1),A(2),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 output through JJJJ=-31966 is summarized below. What follows is a hand copy; immediately below there is no rounding by hand.

0 0 0 -31995
-2 -2 0 -31994
-4 -4 0 -31993
1 1 0 -31992
0 0 0 -31991
0 0 0 -31986
-3 -3 0 -31985
-2 -4 0 -31983
1 1 0 -31982
-4 -2 0 -31979
-2 -4 0 -31978
-4 -2 0 -31973
-3 -3 0 -31972
1 1 0 -31970
-2 -4 0 -31968
0 0 0 -31967
4 2 0 -31966

One notes that the candidate solutions at JJJJ=-31995, -31991, -31986, and -31967 are not acceptable.

The message "Division by zero" came many times on screen, but not on paper; one notes that line 1902 here is 1902 LPRINT A(1),A(2),M,JJJJ.

On an old Pentium 4 2.4 personal computer using the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31966 was four 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] M. Gardner (1979), Mathematical Games, Scientific American, 241 (3), Page 25.

[4] M. Gardner (1979), Mathematical Games, Scientific American, 241 (October), Page 26.

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