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Egyptians and Babylonians must have had some ideas about primes but what are the earliest mentions or comments? Where is the first list of primes?

After Burkert's work on Pythagoreanism an answer about 'unidentified pythagoreans' would not count much. Euclide's proof (about the infinity of primes IX.20) is a landmark in mathematics well known enough to make later writings historically unintersting.

More specifically did early Greek philosophers mention primes and are there possibly different words and/or confusions?

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  • $\begingroup$ Burkert's work discredited attribution of what Pythagoreans did to Pythagoras, but all pre-Euclidean sources, e.g. Aristotle, Eudemus and Plato, point to them as the first to advance "number theory". And of course Euclid got the material of books VII-IX from somewhere. The only prior source I can think of possibly involving primes, and in a very oblique way, would be the list of Pythagorean triples in Plimpton 322. $\endgroup$
    – Conifold
    Commented Feb 2, 2016 at 21:59

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For some references, see :

Prime or incomposite numbers and secondary or composite numbers are distinguished in a fragment by Speusippus based upon works of Philolaus . We are told [ Iamblichus, in Nicom.] that Thymaridas called a prime number rectilinear, the ground being that it can only be set out in one dimension (since the only measure of it, excluding the number itself, is $1$).

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The first preserved list of primes is row (b) of the famous Ishango bone: 11, 13, 17, 19. See my answer to the question When were numbers first used for anything other than counting?

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