Repdigit endings to squares

Over at @onthisdayinmath, Pat highlights a @jamestanton question about squares:

$2^2$ ends with 4 and ${12}^2$ ends with 44. Is there a square than ends 444? How about one that ends 4444?

Pat’s answer (yes to the first – ${38}^2=1444$ is the smallest – and probably not to the second) is correct, but I wanted to dig into why no square ends 4444.

One bit of scaffolding I’m going to call on slightly later (a lemma, if you want a technical term to impress people with) is that no square number ends in 11. The quickest way to prove this is to think modulo 20: the only possible remainders for a square number are 0 (for numbers congruent to 0 or 10), 1 (1, 9, 11 and 19), 4 (2, 8, 12 and 18), 9 (3, 7, 13 and 17), 16 (4, 6, 14 and 16) and 5 (5 and 15). 11 is not in that list, and any positive number ending 11 can be written as $20k+11$ for some positive integer $k$.

With that scaffolding under our belt, and our metaphors mixed like a cocktail, we can prove that no numbers ending …4444 are squares.

Proof. Suppose a number $S$ ending …4444 is a square. $S$ can be written as $10000n+4444$ for some positive integer $n$, or as $4(2500n+1111)$.

If this is a square number, then $2500n+1111$ must also be a square number, but it clearly ends in 11 – which contradicts the assumption that it’s a square. ▪

Apart from zeros – you can have as many as you like of them at the end of a square number – four is the only candidate repdigit ending to a square; a 1, 5 or 9 at the end of a square must be preceded by an even number, while 6 must be preceded by an odd.