Michael A. Bennett, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2; email: bennett@math.ubc.ca

Jamie Mulholland, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2; email: jmulholl@math.ubc.ca

Abstract/Résumé:

We show, if \(p\) is prime, that the equation \(x^n+y^n=2pz^2\) has no solutions in coprime integers \(x\), \(y\) and \(z\) with \(|xy|>1\) and prime \(n>p^{27p^2}\), and, if \(p\ne7\), the equation \(x^n+y^n=pz^2\) has no solutions in coprime integers \(x\), \(y\) and \(z\) with \(|xy|>1\), \(z\) even and prime \(n>p^{3p^2}\).

Nous montrons que, si \(p\) est premier, l’équation \(x^n+y^n=2pz^2\) n’a pas de solution parmi les nombres entiers copremiers \(x\), \(y\), \(z\), avec \(|xy| > 1\) et \(n>p^{27p^2}\) premier. Nous montrons aussi que, si \(p\ne7\), l’équation \(x^n+y^n=pz^2\) n’a pas de solution parmi les nombres entiers copremiers \(x\), \(y\), \(z\), avec \(|xy| >1\), \(z\) pair, et \(n>p^{3p^2}\) premier.

Keywords:

AMS Subject Classification: Higher degree equations; Fermat's equation 11D41

PDF(click to download): On the diophantine equation $x^n+y^n=2^{alpha}pz^2$

D. Poulakis, Department of Mathematics, Aristotle University of Thessaloniki, University Campus, 541 24 Thessaloniki, Greece; email: poulakis@ccf.auth.gr

P.G. Walsh, Department of Mathematics, University of Ottawa. 585 King Edward St., Ottawa, Ontario K1N-6N5; email: gwalsh@mathstat.uottawa.ca

Abstract/Résumé:

Ljunggren proved that for a nonsquare positive integer \(d\), the quartic Diophantine equation \(X^2-dY^4=1\) has at most two solutions in positive integers, and gave precise information on the location of these solutions in the case that two such solutions actually do exist. Inspired by recent work of P. Samuel, we show that in the case that \(d>3\) is prime, there is at most one positive integer solution to \(X^2-dY^4=1\), and that it arises from the fundamental solution of the Pell equation \(X^2-dY^2=1\).

Ljunggren a montré que pour un nombre entier positif de nonsquare \(d\), l’équation \(X^2-dY^4=1\) a au plus deux solutions dans des nombres entiers positifs, et a fourni l’information précise sur l’endroit de ces solutions dans le cas que deux telles solutions réellement existent. Inspirer par les travaux récents de P. Samuel, nous montrons cela dans le cas que \(d>3\) est une nombre premier, il y a au plus une solution positive de nombre entier \(X^2-dY^4=1\), et qu’elle résulte de la solution fondamentale de l’équation de Pell \(X^2-dY^2=1\).

Keywords:

AMS Subject Classification: Higher degree equations; Fermat's equation 11D41

PDF(click to download): A note on the Diophantine equation $x^2-dy^4=1$ with prime discriminant

T. Metsänkylä

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: Higher degree equations; Fermat's equation 11D41

PDF(click to download): Catalan's equation with a quadratic exponent

Z. Cao

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: Diophantine equation, Fermat's quotient

AMS Subject Classification: Higher degree equations; Fermat's equation 11D41

PDF(click to download): The Diophantine equations x4 - y4 = zP and x4 - 1 = dyq

S. Mohit / M.R. Murty

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: Congruences; primitive roots; residue systems, Higher degree equations; Fermat's equation 11A07, 11D41

PDF(click to download): Wieferich primes and Hall's conjecture

C. Helou

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: Power residues; reciprocity, Higher degree equations; Fermat's equation, Cyclotomic extensions 11A15, 11D41, 11R18

PDF(click to download): Proof of a conjecture of Terjanian for regular primes

K. Wu / M. Le

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: Higher degree equations; Fermat's equation 11D41

PDF(click to download): A note on the diophantine equation x4 - y4 = zp