On a conjecture on exponential Diophantine equations

Mihai Cipu (1) , Maurice Mignotte (2)

1 IMAR - "Simion Stoilow" Institute of Mathematics
2 IRMA - Institut de Recherche Mathématique Avancée

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.

Mots clés: exponential equations; linear forms in logarithms; Terai's conjecture

Domaines: Théorie des nombres [math.NT]

Mihai Cipu, Maurice Mignotte. On a conjecture on exponential Diophantine equations. 2008. ⟨hal-00343241⟩