Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures
math.NT, cs.AI
Submitted: 2025-08-08
Updated: 2026-09-16
Comments: 26 pages, 2 tables. v2: corrects the previously best known 5, 6 and 9 term relations (Section 1.2, Table 1) and attributes the floor-function iteration to Abrarov et al. (Section 3.3). Results unchanged
Code: https://github.com/ncw/find_arctan_formulae
License: http://creativecommons.org/licenses/by-nc-nd/4.0/
The gist: Machin-like arctangent relations are classical tools for computing π, with efficiency quantified by the Lehmer measure (λ).
Terminology
Abstract
Machin-like arctangent relations are classical tools for computing π, with efficiency quantified by the Lehmer measure (λ). We present a framework for discovering low-measure relations by coupling the PSLQ integer-relation algorithm with number-theoretic filters derived from the algebraic structure of Gaussian integers, making large scale search tractable. Our search yields new 5 and 6 term relations with record-low Lehmer measures (λ=1.4572, λ=1.3291). We also demonstrate how discovered relations can serve as a basis for generating new, longer formulae through algorithmic extensions. This combined approach of a constrained PSLQ search and algorithmic extension provides a robust method for future explorations.