A conjecture which implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set

Details A conjecture which implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set A conjecture which implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set

2011-09-18

arxiv.org/abs/1109.3826v12

Mathematics

Number Theory

9 pages, Theorem 1 added

Scientific paper

We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n|=<2^(2^(n-1)). The conjecture implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set. We present a MuPAD code whose execution never terminates. If the conjecture is true, then the code sequentially displays all integers n>=2. If the conjecture is false, then the code sequentially displays the integers 2,...,n-1, where n>=4, the value of n is unknown beforehand, and the conjecture is false for all integers m with m>=n.

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