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 ...