Generalized Pell-Fermat equations and Pascal triangle

Daniel Gandolfo (1, 2) , Michel L. Rouleux (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E5 Physique statistique et systèmes complexes

Using Pascal triangle, we give a simple generalization to a well-known problem of S.Ramanujan. Thus we are interested in computing the median of some integer valued distributions, arising naturally when extending arithmetic progressions to progressions of polynomial growth. We show this reduces to equations of Pell-Fermat type of higher order, which admit very few integer solutions , but for which, following S.Ramanujan's original idea, we can always find integer sequences of best approximation, in the Diophantine sense. Our procedure relies much on formal Calculus with Mathematica.

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]

Daniel Gandolfo, Michel L. Rouleux. Generalized Pell-Fermat equations and Pascal triangle. 2021. ⟨hal-02897097v3⟩