[FOM] History of Fundamental Theorem of Arithmetic

joeshipman@aol.com joeshipman at aol.com
Thu Nov 15 00:23:07 EST 2007

Bob Knighten recommended the article "A Historical Survey of the 
Fundamental Theorem of Arithmtic" by A.G. Agargun and E.M. Ozkan 
(Historia Mathematica 28 (2001) 207-214), which answers my questions as 

As I had supposed, the closest Euclid comes to the FT of A is 
Proposition IX.14, but he did not go further because he had no concept 
of factorization, or indeed of a product of a set of more than 3 
numbers (the most Euclid talks about is common multiples of a set of 
numbers, not their product). However, Euclid VII.30 (a prime divides a 
product only if it divides one of the factors) and Euclid VII.31 (any 
composite number is divisible by a prime) together allow anyone who 
actually formulates the FT of A to prove it.

The existence part of the FT of A was clearly stated and proved by the 
Persian Kamal al-Din al-Farisi (who died ca. 1320). He did not state 
uniqueness, but he uses the prime factorization to find all the 
divisors of a number and states that all the divisors of a number are 
given by taking the prime factors in a factorization, products of all 
pairs of prime factors, products of all triples of prime factors, etc. 
It is quite easy to get true uniqueness from this, but not completely 
trivial (you need, in modern terminology, that different multisets 
cannot have the same sub-multisets); however, that step is small enough 
that I am willing to give al-Farisi credit for being the first to state 
the uniqueness part.

In 1689 Jean Prestet also investigated the divisors of a number, and 
gave a statement roughly the same as al-Farisi's (though less 
precisely). In 1770 Euler did the same, and in 1798 Legendre did the 
same. None of them proved it, although Legendre came a bit closer by 
giving a canonical prime factorization (successively dividing by 2 as 
far as possible, then 3, then 5, etc.).

In 1801, Gauss (in Disquisitiones Arithmeticae Article 16) clearly 
states AND PROVES uniqueness. (He stated but did not prove existence 
because it was too easy.) In Clarke's translation:

"It is clear from elementary considerations that any composite number 
can be resolved into prime factors, but it is tacitly supposed and 
generally without proof that this cannot be done in many various ways."

I don't think Gauss would have used "tacitly supposed and generally 
without proof", and then provided a proof, if there had been an actual 
proof in any of the sources available to Gauss. The jump was small, but 
apparently no one before Gauss made it. (Of course, there were many 
jumps that look small to us now that were too big for Gauss to make. It 
is difficult to see things as our distant predecessors did.)

I am now satisfied about the historical status of the FT of Arithmetic 
and of the irrationality of square roots, but I'd still like to know:

1) Who first published a proof that cube roots of integers that are not 
perfect cubes were irrational? (Theaetetus claimed to have proven it 
but does not give the proof.)
2) Who first stated that kth roots of integers that are not perfect kth 
powers were irrational, for all k?
3) Who first proved that kth roots of integers that are not perfect kth 
powers were irrational, for all k?

-- JS
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