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 joro shows below is a consequence and not a proof. Another consequence as mentioned is $\#P\subseteq FP/Poly$.
Is this related to non-existence of multiplication in Presburger arithmetic?
Posted here: https://cstheory.stackexchange.com/questions/54524/nonexistence-of-short-integer-program-sequence-which-generates-squares after a week as I have received no relevant answers other than someone here mentioning consequences whose violation is not known and may not be necessary to refute the existence of such programs. Moderators are also unwilling to remove such answers.
