Boğaziçi Üniversitesi Matematik Konuşmaları
Integers which are(n't) the sum of two cubes
Levent Alpöge
Harvard University, Amerika Birleşik Devletleri
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 work with Manjul Bhargava and Ari Shnidman.)
