📝 SummarXiv

Abstract

We study the modular resolution method using a new tool called a polyserie, introduced by Wildberger N.J. \& Rubine D. in their recent article \cite{wildberger2025hyper}. In the present article we try to prove an equivalence theorem of the existence and the uniqueness of the solution of the equation of the form:\\ $H_{t}^{2}\left(x\right) -H_{t}\left( x\right) +t\equiv 0\ \ \left( \mathrm{mod}\text{ }t^{n}\right) $, by using the same recurrence formula introduced in \cite{wildberger2025hyper} between the Catalan sequence terms: $C_{n+1}=\sum_{k=0}^{n}C_{k}C_{n-k}$. We introduce Wildberger's polynumber sequences, binomial Chu-Vandermonde identity, truncated polyseries and finally modular resolution as application.