Zermelo 1934

pp. 574-581

in: Ernst Zermelo, Set theory, miscellanea / Mengenlehre, varia, Berlin, Springer, 2010

Abstract

Zermelo's article 1934, his only one of purely number-theoretic character, contains two remarks on elementary prime number theory. The first, shorter remark is of particular interest. The so-called fundamental theorem of arithmetic, i.e. unique factorization in ℕ, was first explicitly stated and proved by Carl Friedrich Gauß (Gauß 1801). The proof of the uniqueness rests upon Euclid's Theorem 30 in book VII of the Elements: If a prime p divides a product, then it divides at least one of the factors. This is derived by means of properties of the greatest common divisor and uses Euclid's algorithm. A proof that dispenses with Euclid's algorithm can be found in Edmund Landau's Elementare Zahlentheorie (Landau 1927).