Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures
This work addresses the incremental improvement of computational efficiency for π calculation, primarily benefiting mathematicians and computational scientists.
The researchers tackled the problem of discovering efficient Machin-like arctangent relations for computing π, quantified by the Lehmer measure, and achieved record-low measures of 1.4572 and 1.3291 for 5 and 6 term relations.
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.