On the divisibility properties of integers (I)
On the divisibility properties of sequences of integers (II)
On the division polynomials of elliptic curves.
On the divisor function over Piatetski-Shapiro sequences
Let be an integer part of and be the number of positive divisor of . Inspired by some results of M. Jutila (1987), we prove that for , where is the Euler constant and is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.
On the divisor problem: Moments of Δ(x) over short intervals
On the divisors of
On the dynamics of in a local field
Let be a local field, and where denotes the characteristic of the residue field. We prove that the minimal subsets of the dynamical system are cycles and describe the cycles of this system.
On the Eisenstein series of Hilbert modular groups.
On the elements of prime power order in K₂ of a number field
On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement.
On the entry sum of cyclotomic arrays.
On the enumeration of quintic fields with small discriminant.
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 .
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.