Integers which are(n’t) the sum of two cubes
SEMINARS: NUMBER THEORY
When: February 15, 2023
3:00 pm - 4:00 pm
Where: Science Center 507
Address:
1 Oxford Street, Cambridge, MA 02138, United States
Speaker: Levent Alpöge
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.)