Note on a Fermat-type Diophantine equation
The author extends his previous results [J.\ Number Theory 80, 174-186 (2000; Zbl 0972.11015)] concerning the Diophantine equation \(x^p+y^p=cpz^p\), where \(p\) is an odd prime and the prime factors of \(c\) are of the form \(kp-1\) with \((k,p)=1\). Let \(x,y,z\) be a solution. Then, as shown by the author [op. cit.], \(p\) is irregular. Assume, in particular, that \(p\) divides at least one of the Bernoulli numbers \(B_{p-3},B_{p-5},\dots,B_{p-n}\) with \(n\) odd, \(3 \leq n \leq p-6\). The author shows under some additional assumptions that the integer \({{pcz^p}\over {x+y}}\) does not contain odd prime factors \(q\) satisfying a condition formulated as follows. Let the \(p\)-part of the ideal class group of the \(p\)th cyclotomic field \(\mathbb{Q}(\zeta)\) be decomposed in the usual way into the eigenspaces \(C_p^{(i)}\), \(3 \leq i \leq p-2\). The above assumption on \(p\) then implies that the group \(C= C_p^{(3)}\oplus C_p^{(5)}\oplus \cdots \oplus C_p^{(n)}\) is non-trivial. The primes \(q\) in question are primes having a prime ideal factor in \(\mathbb{Z}[\zeta]\) whose ideal class, up to a \(p\)th power factor, is a non-trivial element of \(C\). The additional assumptions mentioned above can be expressed in the following form: \(C_p^{(i)}\) is trivial for \(p-n-1 \leq i \leq p-2\) and \(2^i \not\equiv 1 \pmod p\) for \(1\leq i \leq n+1\). If \(p\) is large compared to \(n\), i.e., \(p > 14\sqrt{n+1}({{n+1}\over 6})^{n+1}\), then these assumptions are automatically satisfied except for the assumption about \(C_p^{(i)}\) for even values of \(i\). If \(p\) is still larger, \(p > e^{t(n)}\) with \(t(n) = n^{224n^4}\), then this last assumption is also satisfied, as follows from a result by \textit{C. Soulé} [J. Reine Angew. Math. 517, 209-221 (1999; Zbl 1012.11094)].
scientific article; zbMATH DE number 1902890
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