A cta Universitatis Matthiae Belii, series Mathematics

Issue 2016, 25–26, ISSN 1338-7111

Online Edition, http://actamath.savbb.sk

A note on Euclid’s Theo rem concerning

the inﬁnitude of the p rimes

Filip Saidak

Dep artment of Mathematics and Statistics, University of North Car olina,

Gr e ensbor o, NC 27402, U.S.A.

saidak@protonmail.ch

Abstract

W e present another elementary proof of Euclid’s Theorem concerning the inﬁnitude of the prime n umbers.

This proof is “geometric” in nature and it employs very little b eyond the concept of “prop ortion. ”

R e ceive d 21 July 2016

R evise d 25 Octob er 2016

A c cepte d in ﬁnal form 27 Octob er 2016

Publishe d online 20 Novemb er 2016

Communic ate d with Mir oslav Haviar.

Keywo rds Euclid, prime num b ers.

MSC(2010) 11A41.

Euclid’s Theorem ([ 2 ], Bo ok IX, Prop osition 20) establishes the existence of inﬁnitely

man y prime n um b ers. It has b een one of the cornerstones of mathematical thought.

More than a dozen diﬀeren t pro ofs of this result, with many clev er simpliﬁcations and

v arian ts, hav e b een published o v er the past t wo millennia (for lists of proofs and go o d

discussions of their historical relev ance, see [ 1 ], [ 3 ], [ 4 ] and [ 6 ]). A decade ago, in [ 5 ], we

ga v e a short direct proof of Euclid’s Theorem that has receiv ed a surprising amount of

atten tion. Here w e would like to presen t another idea, not quite as simple as the ﬁrst

one, but p erhaps equally fundamen tal. It mak es use of the ancien t concept of prop ortion,

the theory of which was p erfected by Pythagoras, Eudoxus and ﬁnally Euclid himself (a

fact demonstrated by the results summarized in Bo ok V of his Elements [ 2 ]).

W e rephrase the problem slightly . The question we ask is: Why cannot pro ducts of

p o wers of a ﬁnite num b er of primes cov er the entire set N ?

W e inv estigate the factorization geometrically and consider the canonical representa-

tion as an op eration (on exp onents) in tw o dimensions, with single prime p ow ers repre-

sen ting what we will call the “vertical” and their pro ducts the “horizon tal” dimensions.

V ertical Dimension . F or a ﬁxed prime num b er p , and 0 ≤ i ≤ m , there are m + 1

p ositiv e integers that can b e 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

prop ortion ∇ ( p

m

) of these pow ers (up to p

m

) w e not only ha ve ∇ ( p

m

) < 1 , for all m > 1

(as w ell as ∇ ( p

m

) → 0 , as m → ∞ ), but also ∇ ( p

m

) > ∇ ( p

m +1

) , b ecause

m + 1

p

m

>

m + 2

p

m +1

⇐ ⇒ 1 −

1

m + 2

>

1

p

. (1)

Th us, considered vertically , the prop ortions are monotonically decreasing.

Copyrigh t

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