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Acta Universitatis Matthiae Belii, series Mathematics
Issue 2016, 25–26, ISSN 1338-7111
Online Edition, http://actamath.savbb.sk
A note on Euclid’s Theorem concerning
the inﬁnitude of the primes
Filip Saidak
Department of Mathematics and Statistics, University of North Carolina,
Greensboro, NC 27402, U.S.A.
saidak@protonmail.ch
Abstract
We present another elementary proof of Euclid’s Theorem concerning the inﬁnitude of the prime numbers.
This proof is “geometric” in nature and it employs very little beyond the concept of “proportion.
Received 21 July 2016
Revised 25 October 2016
Accepted in ﬁnal form 27 October 2016
Published online 20 November 2016
Communicated with Miroslav Haviar.
Keywords Euclid, prime numbers.
MSC(2010) 11A41.
Euclid’s Theorem ([2], Book IX, Proposition 20) establishes the existence of inﬁnitely
many prime numbers. It has been one of the cornerstones of mathematical thought.
More than a dozen diﬀerent proofs of this result, with many clever simpliﬁcations and
variants, have been published over the past two millennia (for lists of proofs and good
discussions of their historical relevance, see [1], [3], [4] and [6]). A decade ago, in [5], we
gave a short direct proof of Euclid’s Theorem that has received a surprising amount of
attention. Here we would like to present another idea, not quite as simple as the ﬁrst
one, but perhaps equally fundamental. It makes use of the ancient concept of proportion,
the theory of which was perfected by Pythagoras, Eudoxus and ﬁnally Euclid himself (a
fact demonstrated by the results summarized in Book V of his Elements [2]).
We rephrase the problem slightly. The question we ask is: Why cannot products of
powers of a ﬁnite number of primes cover the entire set N?
We investigate the factorization geometrically and consider the canonical representa-
tion as an operation (on exponents) in two dimensions, with single prime powers repre-
senting what we will call the “vertical” and their products the “horizontal” dimensions.
Vertical Dimension. For a ﬁxed prime number p, and 0 i m, there are m + 1
positive integers that can be written in the form p
i
, the largest of which is p
m
. Since,
clearly, m + 1 (1 + 1)
m
= 2
m
p
m
, many integers are not of this form; so for the
proportion (p
m
) of these powers (up to p
m
) we not only have (p
m
) < 1, for all m > 1
(as well as (p
m
) 0, as m ), but also (p
m
) > (p
m+1
), because
m + 1
p
m
>
m + 2
p
m+1
1
1
m + 2
>
1
p
. (1)
Thus, considered vertically, the proportions are monotonically decreasing.
c
2016 Matej Bel University
26 Filip Saidak
Horizontal Dimension. Recall that a function f : N C is called multiplicative, if
f(1) = 1 and f (ab) = f (a)f (b), for all a, b N with gcd(a, b) = 1. A critically important
property of the proportions is their multiplicativity. For all k 2, let us deﬁne
(p
m
1
1
· · · p
m
k
k
) :=
#{n = p
a
1
1
· · · p
a
k
k
: 0 a
j
m
j
, for 1 j k}
p
m
1
1
· · · p
m
k
k
,
then, for all permutations of exponents m
i
, we have
(p
m
1
1
· · · p
m
k
k
) = (p
m
1
1
) · · · (p
m
k
k
) < 1. (2)
In other words, the multiplicativity of implies the horizontal monotonicity.
Combining these two monotonic orthogonal trends is enough to prove the inﬁnitude
of the prime numbers. This is because the vertical dimension is (trivially) inﬁnite, and
(1) implies an ever-increasing sparseness of integers represented by a given prime power;
while from the monotonicity property of (2) it follows that the same will remain true
upon any ﬁnite composition of such powers, and therefore only an inﬁnite horizontal
dimension could possibly compensate for the growing deﬁcit and create a complete cover
of N, guaranteed by the unique factorization theorem.
Acknowledgements
I would like to thank Prof. Peter Zvengrowski (University of Calgary), the referee and
the Editor-in-chief for several useful comments.
References
[1] Dickson, L. E., “History of the theory of numbers. Vol 1, Chapter XVIII, 2nd ed., Chelsea
Publishing, New York, 1992.
[2] Euclid, “Elements. Alexandria, c. 300 BC (for a modern edition, see Heath, T. L., “Thirteen
Books of Euclid’s Elements. Dover, New York, 1956).
[3] Narkiewicz, W., “The development of prime number theory. From Euclid to Hardy and
Littlewood. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000.
[4] Ribenboim, P., “The little book of bigger primes. Springer-Verlag, New York, 2004.
[5] Saidak, F., A new proof of Euclid’s theorem, Amer. Math. Monthly 113 (2006), no. 10,
937–938.
[6] Sierpiński, W., “Elementary theory of numbers”, 2nd Edition, Polish Sci. Pub., Warsaw,
1988.