Ternary arithmetic, factorization, and the class number one problem

Aram Bingham

Published 2020-02-06Version 1

Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with respect to ternary multiplication -- is defined, and it turns out that there are very few 3-primes. They correspond to imaginary quadratic fields $\mathbb{Q}(\sqrt{-n})$, $n>0$, with odd discriminant and whose ring of integers admits unique factorization. We also present algorithms for determining representations of numbers as ternary products, as well as related algorithms for usual primality testing and integer factorization.