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Integral points on the elliptic curve y 2 = x 3 - 4 p 2 x

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

Let p be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve E : y 2 = x 3 - 4 p 2 x . Further, let N ( p ) denote the number of pairs of integral points ( x , ± y ) on E with y > 0 . We prove that if p 17 , then N ( p ) 4 or 1 depending on whether p 1 ( mod 8 ) or p - 1 ( mod 8 ) .

Jacobi symbols, ambiguous ideals, and continued fractions

R. A. Mollin (1998)

Acta Arithmetica

The purpose of this paper is to generalize some seminal results in the literature concerning the interrelationships between Legendre symbols and continued fractions. We introduce the power of ideal theory into the arena. This allows significant improvements over the existing results via the infrastructure of real quadratic fields.

Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita, Takafumi Miyazaki (2012)

Colloquium Mathematicae

Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if b 0 ( m o d 2 r ) and b ± 2 r ( m o d a ) for some non-negative integer r, then the Diophantine equation a x + b y = c z has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).

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