We do not know if $P=NP$ or not or if there is a superfast integer mutiplication algorithm. But I do not think either assumption is necessary to answer this question.
Is there a way to show within an integer program with constant number of variables and constraints of length $poly(\log B)$ (say length $\leq10^{1000000}\log B$), it is not possible for a variable to take exactly all the square values up to a bound $B$?
If we have a sequence of such programs, as $B\rightarrow\infty$, we can get $\#P\subseteq FP/Poly$ and may be even $\#P\subseteq FP$. How can we be sure such a sequence of constant number of integer variable and constraint programs does not exist?
What the answer shows in Nonexistence of short integer program sequence which generates squares is a consequence (again as stated in first sentence in this post may not be needed as this question is much weaker than the $P\neq NP$ question.. that is we may not have a short integer program for Squares but still $P=NP$ or $P\neq NP$ could either way) and not a proof. Another consequence as mentioned is $\#P\subseteq FP/Poly$.
Is this related to non-existence of multiplication in Presburger arithmetic? Or Is this kind of short integer program for squares allowed as far as we know?
