Number Theory: Integers which are(n’t) the sum of two cubes

Fermat identified the integers which are a sum of two squares, integral or rational: they are exactly those integers which have all primes congruent to 3 (mod 4) occurring to an even power in their prime factorization — a condition satisfied by 0% of integers!

What about the integers which are a sum of two cubes? 0% are a sum of two integral cubes, but...

Main Theorem:

1. A positive proportion of integers aren’t the sum of two rational cubes,

2. and also a positive proportion are!

(Joint with Manjul Bhargava and Ari Shnidman.)