It's Dusk Somewhere

One reason why the interaction between addition and multiplication is the interesting part of number theory

Posted at — Jan 18, 2016 by Izzy Meckler

I’ve often heard people say that the interesting part of number theory is the interaction between addition and multiplication, but I only yesterday realized one reason why that is, triggered by convincing myself that $a b = \mathrm{lcm}(a,b)\mathrm{gcd}(a,b)$ (a train of thought triggered by seeing the letters “GCD” graffitied on a wall in Chicago).

The reason is essentially that the multiplicative and additive structures of a ring of “numbers” are individually very well understood. Let’s say “number theory” is about studying unique factorization domains which are finitely generated as an abelian group under addition. Then the additive part is well understood by the classification of finitely generated abelian groups, and the multiplicative part (considered as a monoid) is just isomorphic to the multiplicative monoid on the positive integers (given by adding the exponents in the unique factorization) times the group of units in our UFD. So that’s easy to understand as well.

So, any statement about a ring of numbers only involving one of addition or multiplication is easy to answer, which means the only statements we can’t easily answer (which includes all the “interesting” statements) are ones involving both addition and multiplication.