a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase.

where does this analogy of prime numbers to atoms originate, who was the 1

^{st}to use it?

for starters this page includes the quote by Sautoy from 1998 (M. du Sautoy, "The Music of the Primes", Science Spectra 11, 1998)

It remains unresolved but, if true, the Riemann Hypothesis will go to the heart of what makes so much of mathematics tick: the prime numbers. These indivisible numbers are the atoms of arithmetic.

am thinking that this analogy might be very old, say maybe decades or more, but could it predate even 21^{st} century physics? also looking for other extended comparisons of the two beyond a mere passing sentence.

20th centuryphysics in the question. ofc 20th century physics highly revised the idea/ concept of the atom at the turn of the century with QM physics etc. note: even the greeks/ Leucippus/ Democritus over 2 millenia ago posited existence of physical atoms but afaik the idea of relating primes to atoms did not originate with them, it appears only in "modern" thought. also, am looking for anyearlierrefs by experts than the Sautoy one if its not a long list and it may be difficult to definitively isolate the earliest ref. $\endgroup$ – vzn Apr 23 '16 at 18:27atomfor an irreducible element of an integral domain (i.e., a nonunit that cannot be written as a product of nonunits). Thus, the atoms of $\mathbb{Z}$ are the (positive and negative) primes, as in your question. See the Wikipedia article en.wikipedia.org/wiki/Atomic_domain. This terminology seems popular among commutative algebraists who study factorization theory. A more recent, related invention is the playful term "antimatter domain" for an integral domain with no irreducibles (such as the ring of all algebraic integers). $\endgroup$ – so-called friend Don Apr 24 '16 at 15:522more comments