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On Sat, 5 Aug 2017 01:47 am, Ian Kelly wrote: > Here's a much smaller upper bound: > >>>> n = 2 ** 53 >>>> s = isqrt_newton(n) >>>> n > 9007199254740992 >>>> s > 94906265 >>>> m = (s+1)**2 - 1 >>>> m > 9007199326062755 >>>> isqrt_newton(m) == isqrt_float(m) > False Oooh, interesting. How did you get that? By luck, or do you have some reason for picking (s+1)**2 - 1? I have managed to find an upper bound *almost* as small: 9008783381281320 by running a script for the last 15 or 16 hours, which randomly tests ints between lower and upper bound. If it finds a miss, it reduces the upper bound. That started off very promised, and was extremely fast at first, but as the range of values to check gets tighter, it also becomes harder to find any misses. -- Steve ?Cheer up,? they said, ?things could be worse.? So I cheered up, and sure enough, things got worse.

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