Something's weird here... trying to find the correct version.

Maybe the correct version is this one Peter wrote: Show that there are infinitely many positive integers which cannot be expressed as the sum of three squares. My logic is that we all know that any positive integer number can be represented as a sum of four squers.The case for two squers is easy.So the left one is for three squers.

Peter wrote: Something's weird here... trying to find the correct version. Definitely something wrong. I don't remember the correct version.

Maybe it is "the sum of any number of different squares"? Would that be true?

No it would not (all integers $>128$ are a sum of distinct squares). I tend to the three squares, too.

But 3 squares is still trivial since every $n\equiv7 \pmod 8$ is a counterexample... I think we'll just wipe this one out then.

Maybe four distinct squares?Is it true?

Then the set of powers of $2$ does the trick...