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
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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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