Mathematics – Logic
Scientific paper
2011-05-29
Mathematics
Logic
10 pages. arXiv admin note: text overlap with arXiv:0901.2093, arXiv:1102.4122, arXiv:1011.4103. Theorem 3 added
Scientific paper
Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. If Matiyasevich's conjecture on finite-fold Diophantine representations is true, then for every computable function f:N->N there is a positive integer m(f) such that for each integer n>=m(f) there exists a system S \subseteq E_n which has at least f(n) and at most finitely many solutions in integers x_1,...,x_n. This conclusion contradicts to the author's conjecture on integer arithmetic, which implies that the heights of integer solutions to a Diophantine equation are computably bounded, if these solutions form a finite set.
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