[FOM] Historical queries

Alasdair Urquhart urquhart at cs.toronto.edu
Wed Nov 14 10:50:53 EST 2007

Quoting Joe Shipman:

> 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".

I had a look in Dickson's historical compendium "History of the Theory of
Numbers."  Dickson doesn't even seem to mention the Fundamental Theorem
in its modern form.  However, in Volume I, Chapter XIII, he
discusses lists of primes and factor tables.  He mentions factor
tables of Nicomachus and Boethius, also Fibonacci (1202).
Clearly, the construction of such a table involves the tacit
assumption of uniqueness.  I would think it
likely that the recognition, and implicit assumption of
uniqueness of factorization goes back as far as the Pythagoreans.

There still remain two questions:

1.  Who first stated the uniqueness explicitly, as opposed to
assuming it implicitly?

2.  Who first gave a proof of uniqueness?

I looked in Euler's "Elements of Algebra" from 1770, and can find
no explicit statement of uniqueness, though it is clearly assumed.

So, unless somebody can turn up some evidence beyond that provided
by my own cursory investigations, it would seem that Hardy
and Wright are correct, and the answer to both questions is

Gauss, in Section 16 of the Disquisitiones remarks:

 	It is clear from elementary considerations that any
 	composite number can be resolved into prime factors,
 	but it is often wrongly taken for granted that this
 	cannot be done in several different ways.

and goes on to give a fully rigorous proof of this fact.

Alasdair Urquhart

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