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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.
##### MSC:
 11D41 Higher degree equations; Fermat’s equation 11R18 Cyclotomic extensions 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)
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