Exceptional cases of Terai's conjecture on Diophantine equations (Q616143)

Let \(a,b,c\) be relatively prime positive integers such that \(a^p+b^q=c^r\) for fixed integers \(p,q,r\geq 2\). Terai's conjecture says that \(a^x+b^y=c^z\) in positive integers has only the solution \((x,y,z)=(p,q,r)\) except in the cases \((a,b,c)=(1,2,3), (2,7,3), (2,2^{k-2}-1,2^{k-2}+1)\) with \(k\geq 4\) (up to symmetry) with explicitly known solutions \((x,y,z)\). In this paper, the author studies \(q=r=2\) and obtains results regarding the exceptional cases. For \(q=r=2\) the complete parameterizations of \((a,b,c)\) are known. The method of proof is elementary and uses a result of Scott and Styer to bound the exponent \(Z\) for the equation \(A+B=C^Z\) in terms of the largest squarefree divisor of \(AB\).