Wikipedia:Reference desk/Archives/Mathematics/2021 August 10

= August 10 =

Square repunits
For any base, its repunits of length 0 and 1 are squares (02 and 12). And if b = a2 − 1, the base-b repunit of length 2 equals a2 and so is a square. Other than that, square repunits appear to be rare. The base-2 repunits are the Mersenne numbers. Since Mn ≡ 3 (mod 4) for n > 1 while a2 ≡ 0 or 1 (mod 4), we see that no further squares can be found there. But we have 111113 = 121 = 112 and 11117 = 400 = 202, so square repunits of length greater than one are not necessarily impossible. Are there other non-trivial square repunits? --Lambiam 08:17, 10 August 2021 (UTC)
 * A square repunit of length three would amount to b2+b+1=a2, and it's not too hard to show this is impossible for b>1. In fact the only solutions are a=±1, b=0 or -1. A square repunit of length 4 would give b3+b2+b+1=a2, which is degree 3 and a whole other level of difficulty. But b3+b2+b+1 = (b+1)(b+i)(b-i) so you might be able to attack it using unique factorization over Gausian integers. Longer lengths give Diophantine equations of higher degree. This looks suspiciously similar to Catalan's conjecture. --RDBury (talk) 13:53, 10 August 2021 (UTC)


 * For what it's worth, a brute-force search with both base and length <=200 yielded only the examples you give. (Edit after some optimisation: still true for base <=5000, length <=1000) AndrewWTaylor (talk)