The Story
In 2013, Yitang Zhang - a virtually unknown mathematician working as a lecturer at the University of New Hampshire - submitted a paper that would shake the mathematical world.
"His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N."
No matter how far you go into the deserts of the truly gargantuan prime numbers - no matter how sparse the primes become - you will keep finding prime pairs that differ by less than 70 million.
Why This Matters
The Twin Prime Conjecture states that there are infinitely many pairs of primes that differ by exactly 2. Zhang didn't prove this, but he proved something almost as remarkable: there's a bounded gap between infinitely many prime pairs.
Before Zhang, mathematicians couldn't even prove that the gap between consecutive primes stayed bounded at all - let alone by a specific number. 70 million might seem large, but compared to infinity, it's a miraculous constraint.
The Underdog Story
Zhang couldn't even get an academic job because nobody had heard of him. He worked at a Subway sandwich shop and as an accountant before finally landing a lecturer position. Despite this, he had made one of the truly great breakthroughs of the last 50 years.
Since his proof, other mathematicians (including the Polymath project led by Terence Tao) have reduced the bound from 70 million down to just 246.