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Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?
Apoloniusz Tyszka
The height of a rational number is denoted by and equals provided p/q is written in lowest terms. The height of a rational tuple $(x_1,...,x_n)…
On systems of Diophantine equations with a large number of integer solutions
Apoloniusz Tyszka
Let E_n={x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For each integer n \geq 13, J. Browkin defined a system B_n \subseteq E_n which has exactly b_n solutions in integers…
A common approach to three open problems in number theory
Apoloniusz Tyszka
The following system of equations {x_1 \cdot x_1=x_2, x_2 \cdot x_2=x_3, 2^{2^{x_1}}=x_3, x_4 \cdot x_5=x_2, x_6 \cdot x_7=x_2} has exactly one solution in ({\mathbb N}\{0,1})^7, n…
A hypothetical way to compute an upper bound for the heights of solutions of a Diophantine equation with a finite number of solutions
Apoloniusz Tyszka
Let f(n)=1 if n=1, 2^(2^(n-2)) if n \in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \in {6,7,8,...}. We conjecture that if a system T \subseteq {x_i+1=x_k, x_i \cdot x_j=x_k: i,j,k \…