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Let $a$ and $b$ be two integers not both of which are equal to zero. It is an important and well-known fact that $\text{gcd}(a,b)=ax_{0}+by_{0}$ for some integers $x_{0}$ and $y_{0}$. Even though this result was already implicit in the seventh book of the Elements (remember that it can be derived from the Euclidean algorithm), some authors refer to it as "Bézout's identity"; personally, I like better the way in which Professor E. R. Gentile would call it: the universal suggestion in arithmetic.

Do you happen to know who it was that came up with the proof of the universal suggestion wherein the well-ordering principle of $\mathbb{Z}^{+}$ is applied to the non-empty set $\{ax+by \colon (x,y) \in \mathbb{Z}^{2}\} \cap \mathbb{Z}^{+}$ ?

Please let me thank you in advance for your insightful replies and comments!

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    $\begingroup$ All early authors, from Bachet de Meziriac to Euler and Lagrange derived it by running the Euclidean algorithm backwards, see Bullynck, Modular arithmetic before C.F. Gauss. I do not know who first gave the well-ordering proof, but it would not surprise me if it only happened after Zermelo formulated the principle explicitly. A related use of well-ordering in proving uniqueness of prime factorization goes back to Zermelo himself c. 1912, see Origin of well-ordering proof of uniqueness on Math SE. $\endgroup$
    – Conifold
    Oct 28, 2022 at 5:13
  • $\begingroup$ @Conifold: I love Zermelo's proof of the uniqueness of the prime factorization of the integers! Could it be that the proof of which I speak was first given by a German author that was very familiar with Zermelo's works? $\endgroup$ Oct 28, 2022 at 15:22
  • $\begingroup$ Van der Waerden in his Algebra uses well-ordering (without naming it) to prove that in a Euclidean domain every ideal is principal, and then applies that to the sum of principal ideals to get the Bezout's identity. This two-step (or, at least the first step) may go back to Dedekind, Hasse or Emmy Noether, on whose lectures Algebra is based. At some point, one of them or somebody else cut out the middle and applied well-ordering to the sum directly, perhaps, because Zermelo spelled out the key idea. I'd look into those authors and those named on Math SE. $\endgroup$
    – Conifold
    Oct 29, 2022 at 0:25

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