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
Department of Mathematics and Statistics, University of North Carolina,
Greensboro, NC 27402, U.S.A.
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.
Euclid’s Theorem (, 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 , ,  and ). A decade ago, in , 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 ).
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
, the largest of which is p
clearly, m + 1 ≤ (1 + 1)
, many integers are not of this form; so for the
) of these powers (up to p
) we not only have ∇(p
) < 1, for all m > 1
(as well as ∇(p
) → 0, as m → ∞), but also ∇(p
) > ∇(p
m + 1
m + 2
⇐⇒ 1 −
m + 2
Thus, considered vertically, the proportions are monotonically decreasing.
2016 Matej Bel University