Two divisors of \((n^2+1)/2\) summing up to \(n+1\)
type: Item
instance of: scholarly article
title: Two divisors of \((n^2+1)/2\) summing up to \(n+1\)
zbMATH Open document ID: 1161.11004
author: Mohamed Ayad
author: Florian Luca
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/10812
full work available at URL: article01.pdf
review text: In [Complex Var. Elliptic Equ. 51, No. 2, 143--160 (2006; Zbl 1091.12001)], \textit{M. Ayad} asked to prove that there does not exist an odd prime \(p\) and two positive divisors \(d_1\) and \(d_2\) of \((p^2+1)=2\) such that \(d_1+d_2 = p+1\). In this note, the authors prove the following: Theorem. There does not exist an odd integer \(n > 1\) and two positive divisors \(d_1\) and \(d_2\) of \((n^2 + 1)=2\) such that \(d_1 + d_2 = n+1\). Two applications for this result are given, the first deals with the indecomposability of a specific polynomial (already proved in the paper cited above). The second shows that a certain Diophantine equation can have infinitely many integer solutions.
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 5302614
MaRDI profile type: Publication
OpenAlex ID: W2138723444
cites work: Critical points, critical values of a prime polynomial
cites work: Q4533267
cites work: The Diophantine equation f(x) = g(y)
cites work: On Pillai's Diophantine equation
DOI: 10.5802/JTNB.602
description: scientific article; zbMATH DE number 5302614