What question did this study set out to answer?

To extend Willans' formula beyond its original scope to better understand prime distributions, particularly twin primes.

August 5, 2026American Mathematical Monthly1 citations

Extending Willans’ Formula to Twin Primes and Beyond

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FHFoo Jun Han

Key Points

To extend Willans' formula beyond its original scope to better understand prime distributions, particularly twin primes.
Utilized Wilson's theorem to derive new mathematical relationships among primes.
Developed extensions to Willans' formula incorporating properties of twin primes and other prime types.
Demonstrated new relationships between twin primes and the nth prime according to Willans' formula.
Provided evidence that the extended formula enhances prediction accuracy for prime counts.

Abstract

61 years ago, Willans proposed an explicit formula for the nth prime, p(n) 1. Based on Wilson’s theorem—that (p−1)!+1p∈Z iff p is prime or 1—Willans observed that cos 2(πx)=1 when x∈Z, and lies ...

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Foo Jun Han (Mon,) studied this question.

synapsesocial.com/papers/6a72e79c226790f370656ca9 https://doi.org/https://doi.org/10.1080/00029890.2026.2702816

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Extending Willans’ Formula to Twin Primes and Beyond

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