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I did it with BigInteger through Scala's BigInt wrapper.
What I did to get TLEs from cases 4 and 5 was to convert the BigInteger into a string, forcing calculation for all the digits, rather than only calculating the number of digits. It's only necessary to find a sufficiently accurate base-10 logarithm for the BigInteger.
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Project Euler #57: Square root convergents
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I did it with BigInteger through Scala's BigInt wrapper.
What I did to get TLEs from cases 4 and 5 was to convert the BigInteger into a string, forcing calculation for all the digits, rather than only calculating the number of digits. It's only necessary to find a sufficiently accurate base-10 logarithm for the BigInteger.