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_number

If 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

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