Download Variations in Euclid[n]: The Product of the First n Primes Plus One

Published by the Applied Probability Trust
© Applied Probability Trust 2012
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Variations in Euclid [n]: The Product
of the First n Primes Plus One
JAY L. SCHIFFMAN
We undertake an examination encompassing variations on the product
of the first n primes plus one utilized in Euclid’s classical proof that there
are infinitely many primes. In this article, we seek to secure prime outputs
from these generalizations and determine if we achieve a rich source for
twin prime pairs.
Euclid’s classical proof that the number of positive primes is infinite
In Euclid’s famous proof that there are infinitely many primes, we assume on the contrary that
there are only finitely many primes and achieve a contradiction. Euclid considered the number
M = p 1 p 2 p 3 · · · pn + 1
where p1 , p2 , . . . , pn are supposedly all the prime numbers. Observe that p1 = 2 , p2 = 3 ,
p3 = 5 , and so on, and consider the status of this large integer M . None of the primes
p1 , p2 , . . . , pn divides M , yet M must possess a prime factor, so there must be a prime factor
other than p1 , p2 , . . . , pn . Hence there is no largest prime and the number of primes is infinite.
The primorials, Euclid [n] and Euclid [n −2]
The computer software package MATHEMATICA ® refers to the product of the first n primes
plus one as Euclid [n]. Meanwhile, the product of the first n primes is referred
to as a primorial
(the analog of the factorial for prime numbers) and is denoted by pn # = nk=1 pk . While the
number of primes is infinite, it remains an open problem as to whether there are infinitely many
prime outputs generated by Euclid [n] . Euclid [5] is prime while Euclid [6] is composite:
Euclid [5] = 2 · 3 · 5 · 7 · 11 + 1 = 2311,
a prime number, whereas
Euclid [6] = 2 · 3 · 5 · 7 · 11 · 13 + 1 = 30031 = 59 · 509,
a composite integer. It is known that Euclid [n] yields prime outputs for
n = 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616,
1391, 1613, 2122, 2647, 4413, 13494, 31260, 33237.
One of the more interesting activities for number theorists is to secure twin prime pairs, i.e.
pairs of primes differing by two such as (3, 5) , (5, 7) , and (11, 13). Consider the product of
the first n primes minus one which is Euclid [n] − 2 . It is thus useful to note the prime outputs