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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 1st 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 21st century physics? also looking for other extended comparisons of the two beyond a mere passing sentence.

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    $\begingroup$ This is not an analogy - "indivisible object" is the original meaning of "atom", and primes are multiplicatively indivisible. $\endgroup$ – S. Carnahan Apr 23 '16 at 15:39
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    $\begingroup$ As a matter of fact, atomic physics and number theory are not unrelated: the famous pair correlation conjecture by Montgomery involves the same function as random hermitian matrices used to model the energy levels in heavy atoms. Perhaps this kind of "coincidence" lies in the seemingly weird conception that math is in some sense "timeless physics". I tried to explain a little such a conception in math.stackexchange.com/questions/821881/… $\endgroup$ – Sylvain JULIEN Apr 23 '16 at 15:40
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    $\begingroup$ What is the math analogy of subatomic particles like electrons? What about absolute zero temperature? $\endgroup$ – joro Apr 23 '16 at 16:11
  • $\begingroup$ ps oops that meant to say 20th century physics 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 any earlier refs 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 18:27
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    $\begingroup$ In 1968, P.M. Cohn introduced the term atom for 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 15:52
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For an ancient source regarding the "indivisibility" of prime numbers (but avoiding the term "atom"), see:

[Book 1, XI] the prime and incomposite [...] has received this name because it can be measured only by the number which is first and common to all, unity, and by no other. [...] To be sure, when they are combined with themselves, other numbers might be produced, originating from them as from a fountain and a root, wherefore they are called "prime", because they exist beforehand as the beginnings of the others. For every origin is elementary and incomposite, into which everything is resolved and out of which everything is made, but the origin itself cannot be resolved into anything or constituted out of anything.

See [Book 1, VII] for the definition of number:

Number [arithmos] is limited multitude or a combination of units [monadon] or a flow of quantity made up of units; and the first division of numbers is even and odd.

I'm not familiar with the Greek text; the word atomos is referenced, according to the Index, in 1,III,4, in the context of a quotation from Archytas, and 1,VIII,4-5, both with the meaning of "indivisible": the unit is indivisible.

We have to take into account the fact that the philosophical meaning of atom is "overloaded" with the atomist doctrine, while Nichomacus was a Neopythagorean.

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    $\begingroup$ For completeness: prime and incomposite are called πρῶτον and ἀσύνθετον in the original. $\endgroup$ – Emil Jeřábek Apr 25 '16 at 16:43
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I add this as my contribution, just as a footnote of the answer and comments that were posted. My post isn't an answer for your question, just additional remarks that maybe are interesting in my view, thus you or the professors of this site MathOverflow feel free to comment if this isn't suitable as a contribution.

The online encyclopedia Wikipedia has an article for Primon gas.

About the atoms in physics I add as remark the well-known facts that 1) Georges Lemaître (see from the corresponding Wikipedia Georges Lemaître who was his doctoral advisor, and that he is known also in relation to the Hubble–Lemaître law from the Wikipedia Hubble's law) called to his theory with different words than Big Bang, the linked Wikipedia Georges Lemaître refers these as the last words in first paragraph of the article; and 2) the link nucleocosmogenesis in first paragraph of the Wikipedia George Gamow leads to the page of Wikipedia Nucleosynthesis (see the concise/introductory paragraph, that is the first paragraph, that refers the synthesis of first of those).

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  • $\begingroup$ As companion of my post I add that I know (I didn't read the following article, I know it from an informative point of view via a different source) an article that is in the literature, that is P. Leboeuf, A. G. Monastra and O. Bohigas, The Riemannium, Reg. Chaot. Dyn. 6, 205-210 (2001). $\endgroup$ – user142929 Jul 30 '20 at 14:46
  • $\begingroup$ As aside remark, as extension or curiosity for my post, is that the literature of mathematics also refers sometimes the term prime constellations. The reference that I add here is the article Prime Constellation from the online encyclopedia Wolfram MathWorld. $\endgroup$ – user142929 Aug 18 '20 at 19:06

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