Vol. 63, No. 2, 1976

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Formulas for the next prime

Solomon Wolf Golomb

Vol. 63 (1976), No. 2, 401–404
Abstract

In 1971, J. M. Gandhi showed that if the first n primes, p1,p2,,pn are known, then the next prime, pn+1, is given “explicitly” by the formula:

    t ∑   μ(d)-  1
1 < b(    bd − 1 − b ) < b,
d|Pn
(1)

where b is any positive integer 2, where Pn = p1p2pn, where μ(d) is the Möbius function, and where the unique integer value of t which satisfies the indicated inequalities is in fact pn+1.

In this paper, we obtain of the following formulas for pn+1:

pn+1 = lims→∞{Pn(s)ζ(s) 1}1∕s (2)
pn+1 = lims→∞{Pn(s) ζ1(s)}1∕s (3)
pn+1 = lims→∞{ζ(s) Qn(s)}1∕s (4)
and
pn+1 = lims→∞{1 ζ1(s)Q n(s)}1∕s. (5)

Mathematical Subject Classification
Primary: 10A25, 10A25
Milestones
Received: 13 August 1975
Revised: 19 December 1975
Published: 1 April 1976
Authors
Solomon Wolf Golomb