Exact Computation of the Catalan Number $C(2,050,572,903)$

Abstract

This paper presents a two-phase algorithm for computing exact Catalan numbers at an unprecedented scale. The method is demonstrated by computing $C(n)$ for $n = 2,050,572,903$ yielding a result with a targeted $1,234,567,890$ decimal digits. To circumvent the memory limitations associated with evaluating large factorials, the algorithm operates exclusively in the prime-exponent domain. Phase 1 employs a parallel segmented sieve to enumerate primes up to $2n$ and applies Legendre's formula to determine the precise prime factorization of $C(n)$. The primes are grouped by exponent and serialized to disk. Phase 2 reconstructs the final integer using a memory-efficient balanced product tree with chunking. The algorithm runs on a time complexity of $\Theta(n(\log n)^2)$ bit-operations and a space complexity of $\Theta(n \log n)$ bits. This result represents the largest exact Catalan number computed to date. Performance statistics for a single-machine execution are reported, and verification strategies -- including modular checks and SHA-256 hash validation -- are discussed. The source code and factorization data are provided to ensure reproducibility.