1.6 Attempt at getting the history right

You really shouldn’t trust wikipedia, but I looked up some of the history of the Fundamental Theorem of Arithmetic and verified it as carefully as I could using the books I have.

Lemma 1.9 is basically Proposition 30 of Book VII of Euclid. The proof really is very similar to the one Gallian gives, which is basically the one I’ll give in class. Euclid never proved or even stated Theorem 1.8. The closest he came is Proposition 14 of Book IX, which says that the smallest positive integer \(n\) divisible by primes \(p_1,\ldots , p_r\) is not divisible by any other primes.

Apparently, Theorem 1.8 was first stated by Kama al Din al-Farisi, who lived 1267 – 1319.

Gauss gave a proof of uniqueness in Article 16 of his book Disquisitiones Arithmeticae. Wikipedia mistakenly claims that Gauss’s proof uses modular arithmetic. This isn’t true. What Gauss does is this: he notes that, by Euclid’s Lemma, if \(n {\gt} 1\) is an integer with two distinct prime factorizations, then the set of primes appearing in each factorization must be the same, only the values of the positive exponents can possibly be different. So for example, you could conceivably have \(n = p_1^{a_1}\cdots p_r^{a_r} = p_1^{b_1}\cdots p_r^{b_r}\), with \(b_1 {\gt} a_1\). But then \(A:= n/p_1^{a_1}\) would have two prime factorizations \(A = p_2^{a_2}\cdots p_r^{a_r}\) and \(A = p_1^{b_1 - a_1} p_2^{b_2}\cdots p_r^{b_r}\), and \(p_1\) would appear in the second prime factorization, but not in the first contradicting Euclid’s Lemma.

One last piece of history: The nonstandard proof I gave is given on the Wikipedia page for the Fundamental Theorem of Algebra under the heading “Uniqueness without Euclid’s Lemma". It also seems to have been known for quite a long time. Apparently, it is due to Ernst Zermelo (1871 – 1953), who discovered it sometime around 1900 but didn’t publish it until 1934.