Jsun Yui Wong
The following computer program seeks to solve simultaneously the system of thirty-two Diophantine equations and twenty-two variables taken mainly from page 11381 of Abraham, Sanyal, and Sanglikar [1], from page 252 of Waldschmidt [4], and from page 740, page 744, and page 745 of Perez, Amaya, and Correa [3]. These simultaneous equations are as follows:
X(1)^15 + X(2)^15 = 1088090731
X(2)^14 + X(9)^14 = 268451840
X(1)^13 + X(3)^13 = 1222297448
X(3)^11 + X(10)^11 = 411625181
X(3)^12 + X(8)^12 = 244144721
X(11)+X(12)+X(13)+X(14) = 32
X(15) +X(16)+X(17)+X(18) = 20
X(19)+X(20)+X(21)+X(22) = 26
1+3^X(8) = (2^X(5) )*(5 ^X(6))
X(8)^2 +X(5)^2 +X(6)^2 = 3*X(8)*X(5)*X(6)
X(11) ^2 +X(12)^2+ X(13)^2 + X(14) ^2 =372
X(15) ^2 +X(16)^2+ X(17)^2 + X(18) ^2 = 108
X(19) ^2 +X(20)^2+ X(21)^2 + X(22) ^2 = 204
5^X(5)+5^X(1) = 3^X(2) + 7^X(9)
5^X(6)+5^X(3) = 3^X(10) + 7^X(2)
5^X(3)+5^X(7) = 3^X(10) + 7^X(2)
5^X(1)+5^X(5) = 3^X(2) + 7^X(9)
13^X(4)+7^X(5) = 3^X(6) + 5^X(7)
17^X(4)+7^X(5) = 3^X(6) + 5^X(7)
3^X(5)+5^X(4) +7^X(6) = 11^X(7)
X(1) -X(3)+2* X(5) -X(7)-X(9) =-3
X(2)+(2* X(4))^2 -6* X(6)- X(8)+2* X(10) = 8
X(1) -3* X(2)+4* X(4) +X(6)-6*X(7)+ X(8)-2*X(9) =-16
5* X(1)+2* X(2)-8* X(4)-3* X(5)+4* X(6)+X(7)-X(9) =23
2* X(1) +(X(2)+3* X(4))^3 + (5*X(7))^2-6* X(8)+X(9)-9* X(10) =31
(2* X(1) + X(2))^2+3*X(3)-10* X(5)-( X(6)+3*X(7))^3- X(8)-6*X(9) =27
X(11)^X(4)+ X(12)^ X(5) = X(13)^X(6) + X(14)^X(7)
X(15)^X(1)+ X(16)^ X(5) = X(17)^X(2) + X(18)^X(9)
X(19)^X(5)+ X(20)^ X(4) + X(21)^X(6) = X(22)^X(7)
3* X(1) +2* X(2)-5*X(3)- X(4)^4-2* X(5) + X(6)+4*X(7)-10* X(8)+8*X(9) = -9
X(1) ^2-2*(X(2)+ X(4))^3 + X(5)-3* X(6)-X(7)+4*X(9) +15* X(10) = -24
3* X(1) -(2* X(2))^2+10*X(3)-9* X(4)+3* X(5) + X(6)-2*X(7)- 8* X(8)+12*X(9)-5* X(10) = -25
The second equation through the fifth equation shown above are modifications of four equations on page 11381 of Abraham, Sanyal, and Sanglikar [1, Table 2]. The equation on page 740 of Perez, Amaya, and Correa [3] has been relabelled to become the ninth equation above, and the Markoff equation on page 252 of Waldschmidt [4] has been relabelled to become the tenth equation shown above. The fourteenth equation through the twentieth equation are modifications of
5^X(1)+5^X(2) = 3^X(3) + 7^X(4), 5^X(1)+5^X(2) = 3^X(3) + 7^X(4),
5^X(1)+5^X(2) = 3^X(3) + 7^X(4), 5^X(1)+5^X(2) = 3^X(3) + 7^X(4),
13^X(1)+7^X(2) = 3^X(3) + 5^X(4), 17^X(1)+7^X(2) = 3^X(3) + 5^X(4),
and 3^X(1)+5^X(2) +7^X(3) = 11^X(4) on page 744 of Perez, Amaya, and Correa [3, Table 1], respectively. The sixth last equation, X(11)^X(4)+ X(12)^ X(5) = X(13)^X(6) + X(14)^X(7), an exponential Diophantine equation, is a modification of 17^X(1)+7^X(2) = 3^X(3) + 5^X(4) on page 744 of Perez, Amaya, and Correa [3, Table 1]. Similarly, the fifth last equation, X(15)^X(1)+ X(16)^ X(5) = X(17)^X(2) + X(18)^X(9) is a modification of 5^X(1)+5^X(2) = 3^X(3) + 7^X(4) on page 744 of Perez, Amaya, and Correa [3, Table 1], and the fourth last equation, X(19)^X(5)+ X(20)^ X(4) + X(21)^X(6) = X(22)^X(7),
is a modification of 3^X(1)+ 5^ X(2) + 7^X(3) = 11^X(4)
on page 744 of Perez, Amaya, and Correa [3, Table 1].
One notes line 255, line 265, line 275, line 505, line 508, line 509, line 622, line 623, and line 626.
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 22
112 A(J44)= ( RND *20)
113 NEXT J44
128 FOR I=1 TO 1000
129 FOR KKQQ=1 TO 22
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*22)
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
201 IF ( +1088090731#-X(1)^15# )<0 THEN 1670
210 X(2)= ( +1088090731#-X(1)^15# )^(1#/15#)
215 X(9)= ( +268451840#-X(2)^14# )^(1#/14#)
222 X(3)= ( +1222297448#-X(1)^13# )^(1#/13#)
233 X(10)= ( +411625181#-X(3)^11# )^(1#/11#)
244 X(8)= ( +244144721#-X(3)^12# )^(1#/12#)
255 X(11)=32#-X(12)-X(13)-X(14)
265 X(15)=20#-X(16)-X(17)-X(18)
275 X(19)=26#-X(20)-X(21)-X(22)
369 FOR J44=1 TO 22
370 IF X(J44)<0 THEN 1670
371 NEXT J44
399 N(37)=1#+3#^X(8)-(2#^X(5) )*(5# ^X(6))
401 N(39)=X(8)^2# +X(5)^2# +X(6)^2# -3#*X(8)*X(5)*X(6)
505 N(2)=-372#+X(11) ^2 +X(12)^2+ X(13)^2 + X(14) ^2
508 N(1)=-108#+X(15) ^2 +X(16)^2+ X(17)^2 + X(18) ^2
509 N(0)=-204#+X(19) ^2 +X(20)^2+ X(21)^2 + X(22) ^2
511 N(3)= 5#^X(5)+5#^X(1) -3#^X(2) -7#^X(9)
522 N(5)= 5#^X(6)+5#^X(3) -3#^X(10) -7#^X(2)
533 N(7)= 5#^X(3)+5#^X(7) -3#^X(10) -7#^X(2)
602 N(11)= 5#^X(1)+5#^X(5) -3#^X(2) -7#^X(9)
603 N(15)= 13#^X(4)+7#^X(5) -3#^X(6) -5#^X(7)
604 N(18)= 17#^X(4)+7#^X(5) -3#^X(6) -5#^X(7)
605 N(21)= 3#^X(5)+5#^X(4) +7#^X(6) -11#^X(7)
611 N(41)=3# + X(1) -X(3)+2#* X(5) -X(7)-X(9)
613 N(43)=-8# + X(2)+(2#* X(4))^2# -6#* X(6)- X(8)+2#* X(10)
615 N(45)=16# + X(1) -3#* X(2)+4#* X(4) +X(6)-6#*X(7)+ X(8)-2#*X(9)
617 N(47)= -23# +5#* X(1)+2#* X(2)-8#* X(4)-3#* X(5)+4#* X(6)+X(7)-X(9)
619 N(49)=-31# +2#* X(1) +(X(2)+3#* X(4))^3 + (5#*X(7))^2-6#* X(8)+X(9)-9#* X(10)
621 N(51)=- 27# +(2#* X(1) + X(2))^2#+3#*X(3)-10#* X(5)-( X(6)+3#*X(7))^3#- X(8)-6#*X(9)
622 N(54)= X(11)^X(4)+ X(12)^ X(5) – X(13)^X(6)- X(14)^X(7)
623 N(64)= X(15)^X(1)+ X(16)^ X(5) – X(17)^X(2)- X(18)^X(9)
626 N(66)= X(19)^X(5)+ X(20)^ X(4) + X(21)^X(6)- X(22)^X(7)
634 N(53)=9# +3#* X(1) +2#* X(2)-5#*X(3)- X(4)^4-2#* X(5) + X(6)+4#*X(7)-10#* X(8)+8#*X(9)
635 N(55)=24#+X(1) ^2#-2#*(X(2)+ X(4))^3 + X(5)-3#* X(6)-X(7)+4#*X(9) +15#* X(10)
711 N(69)=25# +3#* X(1) -(2#* X(2))^2+10#*X(3)-9#* X(4)+3#* X(5) + X(6)-2#*X(7)- 8#* X(8)+12#*X(9)-5#* X(10)
922 PD1A=-ABS(N(3))-ABS(N(5))-ABS(N(7))-ABS(N(11))-ABS(N(15))-ABS(N(18))-ABS(N(21))-ABS(N(37))-ABS(N(39))-ABS(N(41))-ABS(N(43))-ABS(N(45))-ABS(N(47))-ABS(N(49)) -ABS(N(51))-ABS(N(53))-ABS(N(54))-ABS(N(55))-ABS(N(69))-ABS(N(2))-ABS(N(64))-ABS(N(1))
925 PD1B=-ABS(N(0))-ABS(N(66))
929 PD1=PD1A+PD1B
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 22
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1468 NN(0)=N(0)
1470 NN(1)=N(1)
1471 NN(2)=N(2)
1481 NN(3)=N(3)
1483 NN(5)=N(5)
1485 NN(7)=N(7)
1491 NN(11)=N(11)
1493 NN(15)=N(15)
1495 NN(18)=N(18)
1501 NN(21)=N(21)
1511 NN(37)=N(37)
1513 NN(39)=N(39)
1514 NN(41)=N(41)
1515 NN(43)=N(43)
1516 NN(45)=N(45)
1517 NN(47)=N(47)
1518 NN(49)=N(49)
1519 NN(51)=N(51)
1521 NN(53)=N(53)
1523 NN(55)=N(55)
1524 NN(69)=N(69)
1526 NN(54)=N(54)
1528 NN(64)=N(64)
1530 NN(66)=N(66)
1557 GOTO 128
1670 NEXT I
1889 IF M<-1 THEN 1999
1901 PRINT A(1),A(2),A(3),A(4),A(5)
1902 PRINT A(6),A(7),A(8),A(9),A(10)
1903 PRINT A(11),A(12),A(13),A(14)
1904 PRINT A(15),A(16),A(17),A(18)
1905 PRINT A(19),A(20),A(21),A(22)
1906 PRINT M,JJJJ
1907 PRINT NN(0),NN(1),NN(2),NN(3),NN(5),NN(7)
1908 PRINT NN(11),NN(15),NN(18)
1909 PRINT NN(21),NN(37)
1910 PRINT NN(39),NN(41),NN(43),NN(45),NN(47)
1911 PRINT NN(49),NN(51),NN(53),NN(54),NN(55),NN(64),NN(66),NN(69)
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=32000 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
3 4 5 0 1
1 1 2 2 6
17 7 3 5
5 5 3 7
8 6 2 10
-1 -9811
0 0 0 0 0
0
0 0 0
0 0
0 0 0 0 0
0 0 0 0 0
0 1 0
3 4 5 0 1
1 1 2 2 6
17 7 5 3
5 5 3 7
6 10 2 8
-1 -2359
0 0 0 0 0
0
0 0 0
0 0
0 0 0 0 0
0 0 0 0 0
0 1 0
3 4 5 0 1
1 1 2 2 6
17 7 5 3
5 5 3 7
8 6 2 10
-1 3030
0 0 0 0 0
0
0 0 0
0 0
0 0 0 0 0
0 0 0 0 0
0 1 0
3 4 5 0 1
1 1 2 2 6
17 7 5 3
5 5 3 7
3 5 7 11
0 22974
0 0 0 0 0
0
0 0 0
0 0
0 0 0 0 0
0 0 0 0 0
0 0 0
3 4 5 0 1
1 1 2 2 6
17 7 3 5
5 5 3 7
6 10 2 8
-1 23834
0 0 0 0 0
0
0 0 0
0 0
0 0 0 0 0
0 0 0 0 0
0 1 0
3 4 5 0 1
1 1 2 2 6
17 7 3 5
5 5 3 7
7 5 3 11
0 29297
0 0 0 0 0
0
0 0 0
0 0
0 0 0 0 0
0 0 0 0 0
0 0 0
During this run from JJJJ=-32000 through JJJJ=32000, “Overflow” was observed about 100 times.
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=32000 was 13 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] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[3] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[4] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.
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