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11D41, Higher degree equations; Fermat's equation — 7 results found.

      
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On the diophantine equation $x^n+y^n=2^{\alpha}pz^2$
C. R. Math. Rep. Acad. Sci. Canada Vol. 28, (1), 2006 pp. 6–11
Michael A. Bennett; Jamie Mulholland (Received: 2006/03/01)

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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.

A note on the Diophantine equation $x^2-dy^4=1$ with prime discriminant
C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (2), 2005 pp. 54–57
D. Poulakis; P.G. Walsh (Received: 2005/02/23)

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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 \(X^2-dY^2=1\).

Catalan’s equation with a quadratic exponent
C. R. Math. Rep. Acad. Sci. Canada Vol. 23 (1) 2001, pp. 28–32
T. Metsänkylä (Received: 2000/11/03)

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No abstract available but the full text is online at the title link above.

The Diophantine equations x4 – y4 = zp and x4 – 1 = dyq
C. R. Math. Rep. Acad. Sci. Canada Vol. 21 (1) 1999, pp. 23–27
Z. Cao (Received: 1998/10/13)

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No abstract available but the full text is online at the title link above.

Wieferich primes and Hall’s conjecture
C. R. Math. Rep. Acad. Sci. Canada Vol. 20 (1) 1998, pp. 29–32
S. Mohit / M.R. Murty (Received: 1997/09/17)

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No abstract available but the full text is online at the title link above.

Proof of a conjecture of Terjanian for regular primes
C. R. Math. Rep. Acad. Sci. Canada Vol. 18 (5) 1996, pp. 193–198
C. Helou (Received: 1996/07/31)

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A note on the diophantine equation x4 – y4 = zp
C. R. Math. Rep. Acad. Sci. Canada Vol. 17 (5) 1995, pp. 197–200
K. Wu / M. Le (Received: 1995/08/11)

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

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05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F11 11F66 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K14 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F25 39B72 42C05 43A07 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 83C05

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