A prime prime primer

0.04 MB·

by  Michael J. Caola

A prime prime primer
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0 Michael Caola
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6 Normanton Rd., Bristol, BS8 2TY, U.K.

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[email protected]
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6 February 8, 2008

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Abstract
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Wepresentasimple,closedformulawhichgivesalltheprimesinorder.
. It is a simple product of integer floor and ceiling functions.
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t
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m 1 Introduction
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Interest in prime numbers continues apace, e.g. computing, secure commu-
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nications, quantum computing, statistical physics, engineering and, of course,
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mathematics. However,wearestillfarfromfindingausefulfunctionp(n)which
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4 generates all the primes in order. We present such a function next and discuss
0 its usefulness [1]. All variables below, save x, are integers.
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4 2 Analysis
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/
h Consider the product function
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a
√n
m ⌊ ⌋ n n
p(n)= − , (1)
v: nY′=2(cid:18)(cid:24)n′(cid:25) (cid:22)n′(cid:23)(cid:19)
i where the integer floor ⌊x⌋ (ceiling ⌈x⌉) functions give the greatest (smallest)
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integer not greater (not smaller) than x: e.g. ⌊7⌋ = ⌈7⌉ = 7, ⌊3.7⌋ = 3,
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a ⌈3.7⌉=4, ⌈−7.3⌉=−7, ⌊−7.3⌋=−8.
Then, for n=2,3,4...,
1, if n is prime
p(n)= (2)
(cid:26) 0, if not
Also,wehavedirectlyfrom(2)thatthenumberπ(n)ofprimesnotexceeding
n is
n
π(n)= p(n′) (3)
nX′=1
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3 Discussion
• That function (1) satisfies (2) may be seen by inspection of ⌈⌉ - ⌊⌋:
0, if x is an integer
⌈x⌉−⌊x⌋= (4)
(cid:26) 1, if not
Thus p(n) = 1 only if all the divisors n of n give non-zero remaninders,
′
which is the definition of primality.
• What constitutes a “useful function” p(n) is partly subjective [1], so we
describe our p(n) as: simple, compact, transparent, and using only two
basic notions (floor ⌊⌋) and (ceiling ⌈⌉) of integer mathematics.
• We have programmedand checked algorithm (1,2,3) on a PC.
Reference
[1] Hardy, G. H. & Wright, E. M. (1978) ’An introduction to the theory of
numbers’, ( 5th edition). Clarendon Press, Oxford.
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