A famous theorem by¹ Fermat is that four squares cannot form an arithmetic progression. (Three definitely can: $1^2 = 1$, $5^2 = 25$ and $7^2 = 49$ already give an example).
It already has been discussed on this website (e.g. here) and has non trivial elementary proofs (e.g. by van der Poorten) as well as more conceptual proofs with modern geometric tools (e.g. in this Conrad blurb).
My question is:
Is there a much easier proof of the weaker statement that five squares cannot form an arithmetic progression?
Feel free to replace five by six, twelve or a billion if it makes your life easier!
¹: Well "by" Fermat in the usual meaning: Fermat boasted to his penpals that he could prove it and Euler and Legendre proved it a century later...
