zbMATH — the first resource for mathematics

A class number criterion for the equation \((x^p-1)/(x-1)=py^q\). (English) Zbl 1136.11024
For an odd prime \(p\), let \(\Phi_p(x)=(x^p-1)/(x-1)\), where \(x\) is a rational integer. For simplicity, we write \(\Phi(x)\) instead of \(\Phi_p(x)\). If \(x\not\equiv 1\pmod{p}\) then \(p\) does not divide \(\Phi(x)\) and if \(x\equiv 1\pmod{p}\) then \(p\) exactly divides \(\Phi(x)\). Now, consider a prime \(q\neq p\). In view of the above remarks, a natural problem is the study of the Diophantine equations \(\Phi(x)=y^q\) (the solutions must satisfy \(x\not\equiv 1\pmod{p}\)) and \(\Phi(x)=py^q\) (the solutions must satisfy \(x\equiv 1\pmod{p}\)).
Recently, these equations have been studied by P. Mihǎilescu [J. Number Theory 118, 123–144 (2006; Zbl 1104.11049) and ibid. 124, No. 2, 380–395 (2007; Zbl 1127.11021)]. The first of these equations is commonly believed to have some well-known solutions and no others. In the present paper the second equation is studied.
According to Theorem 1 of the second aforementioned paper, if \(q\) does not divide \(h_p^{-}\) (the \(p\)th relative class-number) and, in addition, some other quite technical condition is satisfied, then there are no solutions \((x,y)\) in integers \(x,y\neq 1\). The present author manages to remove that technical condition, thus offering us a nice general Diophantine result. As the author says, his work is inspired by Mihǎilescu’s recent work. The case \(q\equiv 1\pmod{p}\) is treated à la Mihǎilescu, while for the case \(q\equiv 1\pmod{p}\) some new ideas have been introduced by the present author.
11D41 Higher degree equations; Fermat’s equation
11R18 Cyclotomic extensions
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
Full Text: DOI