American Math. Mon thly , V ol 113, No. 10, Decem b er 2006.

A NEW PR OOF OF EUCLID’S THEOREM

FILIP SAID AK

A prime n um b er is an in teger greater than 1 that is divisible only

b y 1 and itself. Mathematicians ha v e b een studying primes and their

prop erties for o v er t w en t y -three cen turies. One of the very ﬁrst results

concerning these num bers w as presumably prov ed b y Euclid of Alexan-

dria, sometime b efore 300 B.C. In Bo ok IX of his legendary Elements

(see [2]) w e ﬁnd Prop osition 20, whic h states:

Prop osition . Ther e ar e inﬁnitely many prime numb ers.

Euclid’s pro of (mo dernized) . Assume to the contrary that the set

P of all prime n umbers is ﬁnite, say P = { p

1

, p

2

, · · · , p

k

} for a p ositiv e

in teger k . If Q := ( p

1

p

2

· · · p

k

) + 1, then gcd( Q, p

i

) = 1 for i = 1 , 2 , · · · k .

Therefore Q has to ha ve a prime factor diﬀeren t from all existing primes.

That is a con tradiction. 

T oday man y pro ofs of Euclid’s theorem are kno wn. It ma y come as a

surprise that the follo wing almost trivial argumen t has not b ee n giv en

b efore:

New pro of . Let n b e an arbitrary p ositiv e in teger greater than 1. Since

n and n + 1 are consecutiv e in tegers, they m ust be coprime. Hence the

n umber N

2

= n ( n + 1) m ust ha v e at least t wo diﬀerent prime factors.

Similarly , since the in tegers n ( n + 1) and n ( n + 1) + 1 are consecutiv e, and

therefore coprime, the n umber N

3

= n ( n + 1)[ n ( n + 1) + 1] m ust ha ve

at least three diﬀeren t prime factors. This pro cess can b e con tinued

indeﬁnitely , so the n um b er of primes m ust b e inﬁnite. 

1