Dernier théorème de Fermat et groupes de classes dans certains corps quadratiques imaginaires
Determinants of knots and Diophantine equations
Détermination de courbes elliptiques pour la conjecture de Szpiro
Diophantine equations after Fermat’s last theorem
These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:(i)Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?(ii)Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?
Diophantine equations and class number of imaginary quadratic fields
Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and , and let denote the class number of the imaginary quadratic field . In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then , where D, and n satisfy , x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.
Diophantine equations and class numbers of real quadratic fields
Diophantine equations and identities.
Diophantine equations in cyclotomic fields.
Diophantine equations of the form x2 + D = yn.
Diophantine equations with Euler polynomials
We determine decomposition properties of Euler polynomials and using a strong result relating polynomial decomposition and diophantine equations in two separated variables, we characterize those g(x) ∈ ℚ [x] for which the diophantine equation with k ≥ 7 may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.
Distribution of irregular prime numbers.
Distribution of solutions of diophantine equations , where are polynomials
Distribution of solutions of diophantine equations , where are polynomials. - II
Du premier cas du theoreme de Fermat