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Willans' Formula

Willans' formula is a prime-generating formula due to Willans (1964) that is defined as follows. Let

F(j)=|_cos^2[pi((j-1)!+1)/j]_|
(1)
={1 for j=1 or j prime; 0 otherwise
(2)

for a positive integer j, where |_x_| is the floor function. This formula is a consequence of Wilson's theorem and conceals the prime numbers j as those for which F(j)=1, i.e., the values of F(j) are 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, ... (OEIS A080339). Then, for a positive integer x,

pi(x)=-1+sum_(k=1)^xF(k).
(3)

For a positive integer n, the nth prime number p_n is

p_n=1+sum_(m=1)^(2^n)|_|_n/(sum_(j=1)^(m)F(j))_|^(1/n)_|
(4)
=1+sum_(m=1)^(2^n)|_|_n/(1+pi(m))_|^(1/n)_|,
(5)

where pi(m) is the prime counting function (Willans 1964; Havil 2003, pp. 168-169).

See also

Prime Formulas, Prime Number, Wilson's Theorem

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References

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Rowland, E. "An Exact Formula for the Primes: Willans' Formula." Sept. 22, 2022. https://www.youtube.com/watch?v=j5s0h42GfvM.Sloane, N. J. A. Sequence A080339 in "The On-Line Encyclopedia of Integer Sequences."Willans, C. P. "On Formulae for the nth Prime Number." Math. Gaz. 48, 413-415, 1964. https://doi.org/10.2307/3611701.

Referenced on Wolfram|Alpha

Willans' Formula

Cite this as:

Weisstein, Eric W. "Willans' Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WillansFormula.html

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