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 \(bX^ n+cY^ n=dZ^ n\)
type: Item
instance of: scholarly article
title: 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 \(bX^ n+cY^ n=dZ^ n\)
zbMATH Open document ID: 0531.10017
DOI: 10.1016/0022-314X(84)90040-4
author: Barry J. Powell
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 3841956
zbMATH Keywords: Fermat last theorem
zbMATH Keywords: general Fermat equation
MaRDI profile type: Publication
cites work: Q5623788
cites work: Q3867858
cites work: An extension of Sophie Germain's method to a wide class of diophantine equations.
full work available at URL: 0022-314x(84)90040-4
OpenAlex ID: W2119572784
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description: scientific article; zbMATH DE number 3841956