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We consider the Lebesgue-Ramanujan-Nagell type equation , where and are unknown integers with . 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 -integral points on a class of elliptic curves.
This paper investigates the system of equations
in positive integers , , , , where and are positive integers with . In case of 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 with the condition . Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero .
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 , 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...
We consider the Diophantine equation , 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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