Mihai Cipu and Maurice Mignotte: On Terai's conjecture, p.231-237

Abstract:

We give strong bounds for putative counterexamples to a conjecture of Terai (1994) asserting that if $a$, $b$, $c$ are fixed coprime integers with $\min (a,b,c)>1$ such that $a^2+b^2=c^r$ for a certain odd integer $r>1$, then the equation $a^x+b^y=c^z$ has only one solution in positive integers with $\min (x,y,z)>1$. Moreover, we confirm the conjecture in case $z$ is multiple of 3.

Key Words: Simultaneous exponential equations, linear forms in logarithms.

2000 Mathematics Subject Classification: Primary: 11D09;
Secondary: 11D45, 11J20, 11J86.

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