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Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis
08-23-2017, 12:03 AM
Post: #52
RE: Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis
This is an interesting thread! I don't have much time to try anything, but from what I've read the solutions are focusing on computing cubes of numbers and seeing which ones have the proper digits.
Has anyone tried the opposite way? Look at numbers with the correct digits, discard all the ones that cannot be perfect cubes (by the sum of their digits), and simply take the cube root of the ones left?
Here's the way I would try if I could:
Since only 2,3,5,7 are allowed as digits, a base-4 system could encode all possible digits. So let's use a binary integer to encode the digits, 2 bits per digit. For example with two 64-bit numbers combined into a 128-bit you can have 64 digits. Now all we have to do is count from 1 to 2^128 :-))
Write a routine that (decodes and) adds the digits to discard the ones that are not perfect cubes. There's other possibilities: for example a table with the last 2 or 3 digits can help cross out a few more.
If a number has a chance of being a perfect cube, then and only then we do 1/3*ln(A), then exp(), take the nearest integer, cube it and see if it matches.

I have no idea which percentage of numbers could be discarded this way, so it might be a terrible idea, but the inner loop is just bits, tables and ANDs, so it should be relatively efficient until it finds a candidate.

I just thought looking at it backwards might be worth exploring too.
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RE: Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis - Claudio L. - 08-23-2017 12:03 AM

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