In this paper we study the equation x+7 = y, in integers x, y, m with m ≥ 3, using a Frey curve and Ribet’s level lowering theorem. We adapt some ideas of Kraus to show that there are no solutions to the equation with m prime and 11 ≤ m < 10.

Let p be an odd prime and a, b positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation y2 = x(x + 2apb)(x − 2apb) can be easily reduced to the resolution of the unit equation u+ √ 2v = 1 over Q( √ 2, √ p). The solutions of the latter equation are given by Wildanger’s algorithm.

Let h denote the class number of the quadratic field Q( √−A) for a square free odd integer A> 1, and suppose that n> 2 is an odd integer with (n,h)= 1 and m> 1. In this paper, it is proved that the equation of the title has no solution in positive integers x and y if n has any prime factor congruent to 1 modulo 4. If n has no such factor it is proved that there exists at most one solution with ...

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ± 3(mod  8), then the equation 8 (x) + p (y) = z (2) has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod  8), then the equation has only the solutions (p, x, y, z) = (2 (q) - 1, (1/3)(q + 2), 2, 2 (q) + 1), where q is an odd prime with q ≡ 1(mod  3); (iii) if p ≡ 1(mod  8)...

In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Con-tejean and Devie 9] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm o...

We show, if p is prime, that the equation xn + yn = 2pz2 has no solutions in coprime integers x and y with |xy| ≥ 1 and n > p132p , and, if p 6= 7, the equation xn + yn = pz2 has no solutions in coprime integers x and y with |xy| ≥ 1 and n > p12p .

New results regarding the full solution of the diophantine equationx2+2k=yn in positive integers are obtained. These support a previous conjecture, without providing a complete proof.

We construct a family of Diophantine triples {c1(t), c2(t), c3(t)} such that the elliptic curve over Q(t) induced by this triple, i.e.: y = (c1(t) x + 1)(c2(t) x + 1)(c3(t) x + 1) has torsion group isomorphic to Z/2Z× Z/2Z and rank 5. This represents an improvement of the result of A. Dujella, who showed a family of this kind with rank 4. By specialization we obtain two examples of elliptic cur...

A perfect power is a positive integer of the form ax where a ≥ 1 and x ≥ 2 are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given k ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2, to the Diophantine equation ax − by = k is finite. This conjecture amounts to sayi...

A frequently occurring problem in the theory of binary quadratic forms is to determine, for a given integer m, the existence of solutions to the Diophantine equation f(x, y): = ax + hxy + cy = m, having discriminant A = b 4ac. In the case of a strictly positive nonsquare discriminant, it is well known that the occurrence of one solution to f(x,y) = m implies the existence of infinitely many oth...