Analysis. The proof just given is conceptually even simpler than the
original proof due to Euclid, since it does not use Eudoxus’s method of
“reductio ad absurdum,” proof by contradiction. And unlike most other
proofs of the theorem, it does not require Proposition 30 of Elements
(sometimes called “Euclid’s Lemma”) that states: if p is a prime and
p|ab, then either p|a or p|b. Moreover, our proof is constructive, and it
gives integers with an arbitrary number of prime factors.
Remarks. In Ribenboim [4, pp.3–11] and Narkiewicz [3, pp.1–10] one
ﬁnds at least a dozen diﬀerent proofs of the classical theorem of Euclid,
and many other variations of the arguments listed in , , and  have
been published over the years (in chronological order) by: Goldbach
(1730), Euler (1737 and 1762), Kummer (1878), Perott (1881), Stieltjes
(1890), Thue (1897), Brocard (1915), P´olya (1921), Erd˝os (1938), Bell-
man (1947), F¨urstenberg (1955), Barnes (1976), Washington (1980),
and others. Goldbach’s proof (see , p.4), which uses pairwise copri-
mality of Fermat numbers, seems to be closest in spirit to the argument
we have presented.
ACKNOWLEDGMENTS. Personal and virtual conversations with
Professors Paulo Ribenboim (Queen’s University) and Eduard Kos-
tolansky (Bratislava) are gratefully acknowledged. I would also like to
thank Professor W ladyslaw Narkiewicz (Wroclaw) for bringing to my
attention Hermite’s very simple proof concerning n! + 1.
 M. Aigner and G. M. & Ziegler, Proofs from THE BOOK, Springer-Verlag, Berlin,
 T. L. Heath,The Thirteen Books of Euclid’s Elements, vol. 2, University Press,
Cambridge, 1908; 2nd ed. reprinted by Dover, New York, 1956.
 W. Narkiewicz, The Development of Prime Number Theory, Springer-Verlag, New
 P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, New York,