Science/ pi · mathematics · number-theory · algorithms

New Formulas Push Pi Calculation Efficiency to Record Lows

A new search algorithm found pi formulas more efficient than any known before, though the real-world payoff is mostly bragging rights.

Mathematicians just found more efficient ways to calculate pi - the old-fashioned way.

Machin-like formulas are equations that add up simple arctangent functions to compute pi, a technique dating back to the 1700s. Researchers built a search tool that combines the PSLQ algorithm, which hunts for hidden mathematical relationships between numbers, with filters based on Gaussian integers (complex numbers with whole-number parts) to narrow the search space. That let them scan far more candidate formulas than earlier searches could handle. The result: a five-term formula with a Lehmer measure of 1.4572 and a six-term formula at 1.3291, both the lowest ever recorded for their length, where a lower measure means less arithmetic work per digit of pi produced.

This isn't about breaking pi digit-count records - Chudnovsky-style algorithms already do that faster. What matters here is the search method: constrained PSLQ turns an old, mostly hand-tuned corner of math into something a computer can systematically improve, and the same approach can be extended to hunt for even longer, more efficient formulas.

It's a reminder that centuries-old math problems still have unclaimed territory, if you're willing to throw enough compute at Gaussian integers.

TR

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