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Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah, Anup Das, Azizul Hoque (2024)

Archivum Mathematicum

We consider the Lebesgue-Ramanujan-Nagell type equation x 2 + 5 a 13 b 17 c = 2 m y n , where a , b , c , m 0 , n 3 and x , y 1 are unknown integers with gcd ( x , y ) = 1 . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all S -integral points on a class of elliptic curves.

Congruent numbers with higher exponents

Florian Luca, László Szalay (2006)

Acta Mathematica Universitatis Ostraviensis

This paper investigates the system of equations x 2 + a y m = z 1 2 , x 2 - a y m = z 2 2 in positive integers x , y , z 1 , z 2 , where a and m are positive integers with m 3 . In case of m = 2 we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient a with the condition gcd ( x , z 1 ) = gcd ( x , z 2 ) = gcd ( z 1 , z 2 ) = 1 . Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero a .

Contre-exemples au principe de Hasse pour les courbes de Fermat

Alain Kraus (2016)

Acta Arithmetica

Let p be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ℚ, given by equations a x p + b y p + c z p = 0 , with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent p which are counterexamples to the Hasse principle. This is a consequence of the abc-conjecture if p ≥ 5. Using a cyclotomic approach due to H. Cohen and Chebotarev’s density theorem, we obtain a partial result towards this conjecture, by...

Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski, Tomasz Jędrzejak, Karolina Krawciów (2012)

Colloquium Mathematicae

We consider the Diophantine equation ( x + y ) ( x ² + B x y + y ² ) = D z p , where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).

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