Proof of the impossibility of the Fermat equation \(X^ p+Y^ p=Z^ p\) for special values of \(p\) and of the more general equation \(bX^ n+cY^ n=dZ^ n\) (Q788014)

The author studies the equation of the title under the assumption that \(p\) (a prime) and \(n\) are sufficiently large and, for some integer \(m\), the number \(mp+1\) (resp. \(mn+1)\) is a prime. He proves that if \(3\varphi(m)>m\), then (1) the first equation is unsolvable in integers \(X,Y,Z\) prime to \(p\), and (2) the second equation is unsolvable in non-zero integers \(X,Y,Z\) provided the integers \(b,c,d\) satisfy certain conditions. He notes that \textit{P. Ribenboim} has proved the former result with the weaker restriction \(3\nmid m\) in place of \(3\varphi(m)>m\) [J. Reine Angew. Math. 356, 49--66 (1985; Zbl 0546.10013)]. The author's proof is quite elementary. It also leads to statements concerning the equations \(X^{p^ t}+Y^{p^ t}=Z^{p^ t}\) and \(bX^{p^ t}+cY^{p^ t}=dZ^{p^ t},\) where \(t\) is a positive integer and \(p\) is any odd prime such that (*) \(3\varphi(p-1)>p-1\). In fact, the result is that the former equation, say, has no solutions \(X,Y,Z\) prime to \(p\) if \(t\) is sufficiently large. This is a classical theorem of Maillet, even without the restriction (*), but the known proofs are deeper than the present one.