We consider the diophantine equation(*) xp - x = yq - y in integers (x, p, y, q). We prove that for given p and q with 2 ≤ p < q, (*) has only finitely many solutions. Assuming the abc-conjecture we can prove that p and q are bounded. In the special case p = 2 and y a prime power we are able to solve (*) completely.
We discuss the equation in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and α is a positive integer. The technique used to prove Fermat’s Last Theorem shows that the equation has no solutions with α < 1 or b even. When α=1 and b is odd, there are the two trivial solutions (±1, ∓ 1, ±1). In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for p≡ 1 mod 4.
We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.
For any positive integer which is not a square, let be the least positive integer solution of the Pell equation and let denote the class number of binary quadratic primitive forms of discriminant . If satisfies and , then is called a singular number. In this paper, we prove that if is a positive integer solution of the equation with , then maximum and both , are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions .
Skolem conjectured that the "power sum" A(n) = λ₁α₁ⁿ + ⋯ + λₘαₘⁿ satisfies a certain local-global principle. We prove this conjecture in the case when the multiplicative group generated by α₁,...,αₘ is of rank 1.
This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal...