Find the smallest pythagorean triple a<b<c where c/a is within 0.005 of 1.54 and then do the same for c/b.
I saw Charlie's program and thought there had to be a more efficient way to search for the triplets.
I had an idea of incrementing either the leg or hypotenuse depending if the c/a ratio was either high or low. I used the parameterization a=u^2-v^2, b=2*u*v, c=u^2+v^2. If the c/a ratio is too low then v is increased and if the ratio is too high then u is increased.
5 print=print+"output.txt"
10 U=2:V=1:C=5:A=3:R=5/3:X=0
15 D=abs(R-1.54)
20 while X<1000
30 if R>1.54 then U=U+1 else V=V+1
40 C=U^2+V^2:A=U^2-V^2:R=C/A:X=X+1:B=2*U*V
45 D2=abs(R-1.54)
46 if D<D2 then 60
47 D=D2
50 print U;V,C;A;B,R
60 wend:'while x
65 print=print
The original UBASIC program that printed all results is just the multiples of 10. The other lines add record keeping to only print successively better triangles. Lines 5 and 65 are for printing to a file. The output of this program is:
4 2 20 12 16 1.6666666666666666666
6 3 45 27 36 1.6666666666666666666
7 3 58 40 42 1.45
9 4 97 65 72 1.4923076923076923076
11 5 146 96 110 1.5208333333333333333
13 6 205 133 156 1.5413533834586466165
26 12 820 532 624 1.5413533834586466165
39 18 1845 1197 1404 1.5413533834586466165
52 24 3280 2128 2496 1.5413533834586466165
65 30 5125 3325 3900 1.5413533834586466165
78 36 7380 4788 5616 1.5413533834586466165
89 41 9602 6240 7298 1.538782051282051282
102 47 12613 8195 9588 1.5391092129347162903
115 53 16034 10416 12190 1.5393625192012288786
128 59 19865 12903 15104 1.5395644423777416104
141 65 24106 15656 18330 1.5397291773122125702
154 71 28757 18675 21868 1.5398661311914323962
167 77 33818 21960 25718 1.5399817850637522768
334 154 135272 87840 102872 1.5399817850637522768
501 231 304362 197640 231462 1.5399817850637522768
514 237 320365 208027 243636 1.5400164401736313074
681 314 562357 365165 427668 1.5400079416154341188
The last line is impressively close to 1.54 for running less than 5 seconds.
After letting the program run longer (x<100000) the best it found was {u=8801, v=4058, c=93924965, a=60990237, b=71428916, r=1.5400000003279213359}.
Edited on July 9, 2017, 12:11 am
Another approach
