Greg Egan on Nostr: In base 10, there’s an integer whose reciprocal has a repeating block with all 10 ...
In base 10, there’s an integer whose reciprocal has a repeating block with all 10 digits exactly once:
1/72,728 = 0.000(0137498625) ...
I found examples for all even bases from 2 to 34 except 8, 16 and 32. And for a few days I thought “If these are the only examples, surely it can’t be hard to prove there are no others.”
But then I stepped back and looked at the simpler things in number theory people have been trying to prove for centuries.
For example: are 3, 5, 17, 257 and 65537 the only primes of the form 2^(2^n)+1?
https://en.wikipedia.org/wiki/Fermat_numberIf Euler couldn’t prove this, and nobody since has managed to do it either, it seems very unlikely that I’ll ever prove a conjecture about the bases in which the reciprocal of an integer has a non-redundant pandigital reptend.
But check out base 34!
https://gregegan.net/SCIENCE/Reptends/Reptends.html 
Published at
2025-05-04 14:14:55Event JSON
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"content": "In base 10, there’s an integer whose reciprocal has a repeating block with all 10 digits exactly once:\n\n1/72,728 = 0.000(0137498625) ...\n\nI found examples for all even bases from 2 to 34 except 8, 16 and 32. And for a few days I thought “If these are the only examples, surely it can’t be hard to prove there are no others.”\n\nBut then I stepped back and looked at the simpler things in number theory people have been trying to prove for centuries.\n\nFor example: are 3, 5, 17, 257 and 65537 the only primes of the form 2^(2^n)+1?\n\nhttps://en.wikipedia.org/wiki/Fermat_number\n\nIf Euler couldn’t prove this, and nobody since has managed to do it either, it seems very unlikely that I’ll ever prove a conjecture about the bases in which the reciprocal of an integer has a non-redundant pandigital reptend.\n\nBut check out base 34!\n\nhttps://gregegan.net/SCIENCE/Reptends/Reptends.html\nhttps://media.mathstodon.xyz/media_attachments/files/114/449/976/483/147/941/original/b6d0d08b7ea93622.png\n",
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