Number Theory


A Set of Formulas for Primes

Authors: Simon Plouffe

In 1947, W. H. Mills published a paper describing a formula that gives primes : if A = 1.3063778838630806904686144926… then ⌊A^(3^n ) ⌋ is always prime, here ⌊x⌋ is the integral part of x. Later in 1951, E. M. Wright published another formula, if g_0=α = 1.9287800… and g_(n+1)=2^(g_n ) then ⌊g_n ⌋= ⌊2^(…2^(2^α ) ) ⌋ is always prime. The primes are uniquely determined by α , the prime sequence is 3, 13, 16381, … The growth rate of these functions is very high since the fourth term of Wright formula is a 4932 digit prime and the 8’th prime of Mills formula is a 762 digit prime. A new set of formulas is presented here, giving an arbitrary number of primes minimizing the growth rate. The first one is : if a_0=43.8046877158…and a_(n+1)= 〖〖〖(a〗_n〗^(5/4))〗^n , then if S(n) is the rounded values of a_n, S(n) = 113,367,102217,1827697,67201679,6084503671, …. Other exponents can also give primes like 11/10, or 101/100. If a_0 is well chosen then it is conjectured that any exponent > 1 can also give an arbitrary series of primes. The method for obtaining the formulas is explained. All results are empirical.

Comments: 7 Pages.

Download: PDF

Submission history

[v1] 2018-12-31 09:51:52
[v2] 2018-12-31 12:31:57
[v3] 2019-01-07 23:07:56

Unique-IP document downloads: 84 times is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus