[FOM] Historical queries

[joeshipman@aol.com]

Re X1:

Tait says that the Fundamental Theorem of Arithmetic is in Euclid book 
VII. But Euclid only proves the key lemma (Proposition 30) that if a 
prime divides a product then it divides one of the factors. He does not 
appear to have a concept of exponent or power, so he can only talk 
about which primes divide something but not about the existence of 
"factorizations" which involve primes taken multiple times.

The closest Euclid appears to come to the theorem is book IX 
proposition 14, where he states that the least multiple of a set of 
primes is not divisible by any other prime. But the concept of a 
factorization seems to be absent, so that he doesn't really distinguish 
between numbers divisible by the same set of primes except when talking 
about the least of them. Euclid does not even say that the least 
multiple of a set of primes is the product of those primes, that I can 
see. It's conceivable on a first reading of Euclid that a number could 
have two different factorizations which happen to involve the same 
primes taken to different powers. Those of us with a modern 
mathematical education can take the obvious steps to get from VII.30 to 
the Fundamental Theorem of Arithmetic, but Euclid only got as far as 
IX.14.

Urquhart quotes Hardy and Wright that Gauss was the first to state the 
theorem in its modern form, but surely Fermat and Euler must have 
"known" the theorem in the sense in which we know it, since they dealt 
with factorizations as objects. However, one could maintain that, 
although they could have easily proven uniqueness if it had occurred to 
them to, it never occurred to them that it needed proving. The question 
is, did anyone before Gauss state (or prove) a theorem in which there 
is a recognizable notion of "uniqueness of factorization".

Re X3:

Aitken cites Euclid VIII.8, and he derives the corollary that if N has 
a rational kth root, then it has an integer kth root. But Euclid does 
not draw this corollary even though he arguably had the language to do 
so (he talks about "irrational numbers" and in X.9 he DOES prove that 
the square root of a non-square number is irrational). He does not even 
talk about cube roots, although elsewhere he discusses volumes as 
triple products. So it would still be nice to know who first made a 
general statement about the irrationality of roots of non-powers.

Re X2:

Although Euclid did this, the question is whether earlier authors knew 
the general proposition. As Aitken and Tait point out, according to 
Plato's "Theaetetus", Theodorus appears to have gone case by case up to 
17, so he presumably did not have Euclid's general proof. Theaetetus 
states that he himself and "Socrates" (not Plato's teacher Socrates who 
participated in the dialogue, but his "namesake") generalized the 
theorem to the case of arbitrary non-square integers, and then says 
"and the same about solids". So this answers my question regarding X2 
even though we don't have Theaetetus's proof, just Euclid's.  But I 
still don't know what is the first surviving written proof either for 
"solids" or for kth roots, since Euclid did not actually go beyond 
square roots.

In a forthcoming paper cowritten with John H. Conway, I show that, for 
any non-square integer N, there is a very simple geometric proof that 
the square root of N is irrational that could have been demonstrated by 
Theodorus; however, to find the proof requires finding a number of the 
form (k^2)+1 or (k^2)-1 that is divisible by N, and the guarantee that 
such a number exists depends on the theory of the Fermat-Pell equation, 
which no surviving manuscripts indicate was known by the ancients 
(although Diophantus and Archimedes may well have known this fact 
anyway).

-- JS

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