Pell's equation and series expansions for irrational numbers

TL;DR

This paper explores series expansions for square roots of prime numbers using solutions to Pell's equation and hypergeometric identities, resulting in numerous rapidly converging series for these irrationals.

Contribution

It introduces new fast convergent series expansions for square roots of primes based on Pell's equation and hypergeometric series identities.

Findings

Established numerous fast convergent series for p.

Connected solutions of Pell's equation with hypergeometric series.

Provided practical series for efficient computation of p.

Abstract

Solutions of Pell's equation and hypergeometric series identities are used to study series expansions for $\sqrt{p}$ where $p$ are arbitrary prime numbers. Numerous fast convergent series expansions for this family of irrational numbers are established.