Photo: Mochizuki

Whenever a major leap is made in pure mathematics, the question always leads to “Cool, but what does this mean for *me*?”Here’s what you want to know.

Shinichi Mochizuki is a number theorist at Kyoto University.

He went to Philip Exeter Academy, one of the most prestigious High Schools in the country, and graduated in a brief two years. He entered Princeton University at age sixteen and left with a Ph. D at 22. He was a full professor by 33, an absurdly young age for academia. And now, this mathematical rock star may have just cracked one of the most important problems in his field.

In short, the abc conjecture — proposed in 1985 — explores the relationships between prime numbers.

It’s been described as the most important unsolved problem in Diophantine Analysis, a branch of mathematics that — by working with some of the most simple number systems (like ax + by = 1 or xn + yn = zn) explores some of the deepest relationships in maths.

So if you’re looking for an instant “real world application,” hit the back button — but if you want to see why one equation can tell us so much about how numbers work, read on.

The abc conjecture is as follows.

Take three positive integers that have no common factor and where a + b = c. For instance, 5, 8, and 13.

Now take the distinct prime factors of these integers—in this case 2, 5, and 13—and multiply them to get a new number, d.

In most cases, like this one, d is larger than c. The conjecture states that in rare instances where d is smaller than c, it is usually very close to c. The conjecture also shows that there are a finite number of instances where d is smaller than c.

Mochizuki claims to have cracked this conjecture in a 500-page proof.

Even if you didn’t catch all of that, solving this would be the necessary missing link for a dozen different, more advanced problems in Diophantine Analysis, including the legendary mathematical problem *Fermat’s Last Theorem*. If Mochizuki did successfully prove it, he may have singlehandedy moved his field forward two decades.

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