Modular resolutions by polyseries

TL;DR

This paper introduces a novel modular resolution approach utilizing polynumbers and polyseries, establishing an equivalence theorem for solutions of modular quadratic equations through combinatorial and algebraic tools.

Contribution

It presents new mathematical tools and proves an equivalence theorem linking existence and uniqueness of solutions in modular quadratic equations.

Findings

Established an equivalence theorem for modular quadratic solutions.

Connected polynumber sequences with Catalan numbers and binomial identities.

Developed new methods for modular resolution using polyseries.

Abstract

We study the modular resolution method using new tools called polynumbers and polyseries, introduced by Prof. Wildberger N.J. We try to prove an equivalence theorem of the existence and the uniqueness of the solutions of the modular quadratic equations, using the recurrence formula between the Catalan sequence terms and introducing the following notions: Wildberger's polynumber sequences (polynomials), binomial Chu-Vandermonde identity and truncated polyseries.