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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 M_n ≡ 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)[reply]

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)[reply]

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)