Factoring Catalan numbers

TL;DR

This paper presents a novel prime factorization method for Catalan numbers, utilizing layered structures and Chebyshev's segments, and offers software for factorization up to index 10^8.

Contribution

It introduces a new prime factorization approach for Catalan numbers based on layered segments and modified Kummer's theorem, enabling efficient factorization of large indices.

Findings

Prime factors of Catalan numbers are distributed in layers according to Legendre's formula.

A network of Chebyshev's segments is used to organize prime factors within each layer.

Software implementation allows factorization of Catalan numbers with index up to 10^8.

Abstract

The paper describes a prime factorization of the Catalan numbers. Odd prime factors are distributed in layers in accordance with Legendre's formula. The content of each layer is a network of the intervals, Chebyshev's Segments. The primes of Segment are not calculated and are selected on the basis of its bounds. Layers contain non-repeated primes. Repeated factors are formed when primes are duplicated among different layers. The paper slightly modifies Kummer's theorem for the selection of individual prime factors, also starting from the boundaries of Segments. In conclusion, the reader is offered a software service for factorization of the Catalan number with index up to $10^8$