Jsun Yui Wong

The computer program listed below seeks to solve the following equation taken from Rice [16, p. 385]:

20/Y^15+36/Y^25+40/Y^33+475/Y^40-1.12*(Y^40-1)/(X(1)*Y^40)-6/Y^4-3/Y^8-4.5 =0,

where Y=1+X(1)

0 REM DEFDBL A-Z

1 DEFINT I,J,K

2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),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 1

112 A(J44)= RND

113 NEXT J44

128 FOR I=1 TO 1000

129 FOR KKQQ=1 TO 1

130 X(KKQQ)=A(KKQQ)

131 NEXT KKQQ

139 FOR IPP=1 TO FIX(1+RND*0)

140 B=1+FIX(RND*1)

150 R=(1-RND*2)*A(B)

155 REM 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

303 IF X(1)>1# THEN 1670

306 IF X(1)<0# THEN 1670

312 Y=1+X(1)

313 N(7)= 20/Y^15+36/Y^25+40/Y^33+475/Y^40-1.12*(Y^40-1)/(X(1)*Y^40)-6/Y^4-3/Y^8-4.5

322 PD1=-ABS(N(7))

1111 IF PD1<=M THEN 1670

1452 M=PD1

1454 FOR KLX=1 TO 1

1455 A(KLX)=X(KLX)

1456 NEXT KLX

1557 GOTO 128

1670 NEXT I

1889 IF M<-.1 THEN 1999

1917 PRINT A(1),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=-31998 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

9.864729E-02 -4.768372E-07 -32000

9.864729E-02 -4.768372E-07 -31999

9.864726E-02 -2.384186E-06 -31998

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=-31998 was one second.

**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.

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[16} John R. Rice, *Numerical Method, Software, and Analysis*, Secondn Edition. Boston: Academic Press, 1993.

[17] 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.

[18] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[19] A. Quarteroni, R. Sacco, F. Saleri, *Numerical Mathematics*, Second Edition. Berlin: Springer, 2007 Copyright.

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

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

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

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