Jan Kristian Haugland wrote: > > EvilGenius wrote: > > > A couple of days ago I asked myself whether there is any (small) > > number that can be factored into two different pairs of primes. Put > > differently, do any two pairs of primes when multiplied produce the > > same number as product? > > No. The factorization of a positive integer into prime numbers > is unique up to the order of the factors, and this is a result > that dates back to Euclid (500 B.C.?). It is not always the > case in other rings. For example, in the ring Z[sqrt(-5)], the > number 6 can be factorized as both 2 * 3 and > (1 - sqrt(-5)) * (1 + sqrt(-5)).
Do you mean that when we write of factoring a number, we need to specify what structure we are working in? And if some long discussion about factorization starts off using one structure and then moves on to using another, we need to make the change explicit and take care that whatever we need to be true in both structures has been preserved?
No, I can't believe it. Mathematics would be hugely simplified if "factor" just meant whatever one wanted, when one wanted it. Forget about the structure, that is just so much extra baggage.