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The equation x 2 n + y 2 n = z 5

Michael A. Bennett (2006)

Journal de Théorie des Nombres de Bordeaux

We show that the Diophantine equation of the title has, for n > 1 , no solution in coprime nonzero integers x , y and z . Our proof relies upon Frey curves and related results on the modularity of Galois representations.

The exceptional set for Diophantine inequality with unlike powers of prime variables

Wenxu Ge, Feng Zhao (2018)

Czechoslovak Mathematical Journal

Suppose that λ 1 , λ 2 , λ 3 , λ 4 are nonzero real numbers, not all negative, δ > 0 , 𝒱 is a well-spaced set, and the ratio λ 1 / λ 2 is algebraic and irrational. Denote by E ( 𝒱 , N , δ ) the number of v 𝒱 with v N such that the inequality | λ 1 p 1 2 + λ 2 p 2 3 + λ 3 p 3 4 + λ 4 p 4 5 - v | < v - δ has no solution in primes p 1 , p 2 , p 3 , p 4 . We show that E ( 𝒱 , N , δ ) N 1 + 2 δ - 1 / 72 + ε for any ε > 0 .

The integral points on elliptic curves y 2 = x 3 + ( 36 n 2 - 9 ) x - 2 ( 36 n 2 - 5 )

Hai Yang, Ruiqin Fu (2013)

Czechoslovak Mathematical Journal

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6 n 2 - 1 and 12 n 2 + 1 are odd primes, then the general elliptic curve y 2 = x 3 + ( 36 n 2 - 9 ) x - 2 ( 36 n 2 - 5 ) has only the integral point ( x , y ) = ( 2 , 0 ) . By this result we can get that the above elliptic curve has only the trivial integral point for n = 3 , 13 , 17 etc. Thus it can be seen that the elliptic curve y 2 = x 3 + 27 x - 62 really is an unusual elliptic curve which has large integral points.

The intersection of a curve with algebraic subgroups in a product of elliptic curves

Evelina Viada (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider an irreducible curve 𝒞 in E n , where E is an elliptic curve and 𝒞 and E are both defined over ¯ . Assuming that 𝒞 is not contained in any translate of a proper algebraic subgroup of E n , we show that the points of the union 𝒞 A ( ¯ ) , where A ranges over all proper algebraic subgroups of E n , form a set of bounded canonical height. Furthermore, if E has Complex Multiplication then the set 𝒞 A ( ¯ ) , for A ranging over all algebraic subgroups of E n of codimension at least 2 , is finite. If E has no Complex Multiplication...

The Ljunggren equation revisited

Konstantinos A. Draziotis (2007)

Colloquium Mathematicae

We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.

The method of infinite ascent applied on A 4 ± n B 3 = C 2

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

Each of the Diophantine equations A 4 ± n B 3 = C 2 has an infinite number of integral solutions ( A , B , C ) for any positive integer n . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when A , B and C are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions ( A , B , C ) of the Diophantine equation a A 3 + c B 3 = C 2 for any co-prime integer pair ( a , c ) .

Currently displaying 61 – 80 of 148