On the dispersions of the polynomial maps over finite fields.
On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement.
On the equation
On the equation
On the equation . (Sur l'équation .)
On the equation
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.
On the equation
On the equation a³ + b³ⁿ = c²
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.
On the equation a(x...-1)/(x-1)=b(y...-1)/(y-1).
On the equation aX⁴-bY² = 2
On the equation
On the equation with 0 < i₀ < k-1
On the equation .
On the equation x(x+1)... (x+k-1) = y(y+d)... (y+(mk-1)d), m=1,2
On the equation
On the equation
On the exponential diophantine equation
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 .
On the exponential Diophantine equations of the form .
On the exponential local-global principle
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.
On the extendibility of the Diophantine triple .