Some conjectures on integer arithmetic
Apoloniusz Tyszka · arXiv (Cornell University) · 2009
We conjecture: if integers x_1,...,x_n satisfy (x_1)^2>2^(2^n) \vee ...\vee (x_n)^2>2^(2^n), then (\forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)) \wedge (\forall i,j,k \in {1,...,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k)) for some integers y_1,...,y_n satisfying (y_1)^2+...+(y_n)^2>n \cdot ((x_1)^2+...+(x_n)^2). By the conjecture, for Diophantine equations with finitely many integer solutions, the modulus of solutions are bounded by a computable function of the degree and the coefficients of the equation. If the set {(u,2^u): u \in {1,2,3,...}} \subseteq Z^2 has a finite-fold Diophantine representation, then the conjecture fails for sufficiently large values of n.