# Largest Math Proof In The World Solved In 2 Days, 200 Terabytes In Size

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You think writing proofs for algebra during high school was hard? Think again.

A trio of brilliant mathematicians just solved a decade-old puzzle and consequently produced the world's largest mathematical proof — a text file that is a whopping 200 terabytes in size.

How big is it? A single terabyte is so huge that it can hold about 337,920 copies of Leo Tolstoy's "War and Peace," which is one of the longest and thickest novels ever written.

What's more, the mathematical proof's size is actually equivalent to the entire digitized text archive kept at the United States Library of Congress.

The proof also easily smashed the previous record for the world's largest math proof, which was at 13 gigabytes.

So how did the trio of scientists accomplish this feat?

Researchers Marijn Heule of University of Texas, Oliver Kullmann of Swansea University and Victor Marek of University of Kentucky created an outline and programmed a supercomputer to grind through trillions of color combination possibilities to solve the Boolean Pythagorean Triples problem.

The problem, which has puzzled mathematicians since the 1980s, is about Pythagoras' famous theorem related to the length of the sides of a triangle: a2 + b2 = c2.

Certain sets of numbers can satisfy the theorem with whole integers. One example: 32 + 42 = 52.

But what if all integers had to pick a color: red or blue?

And so the Boolean Pythagorean Triples problem asks: is it possible to color each positive integer either blue or red such that no set of integers a, b and c were all the same color?

To solve this puzzle, Heule, Kullmann and Marek applied the paradigm known as Cube-and-Conquer — a hybrid of the SAT method for difficult problems. The paradigm uses both CDCL solvers and look-ahead techniques.

Before programming the proof into the supercomputer, the trio also did math on their own by using several techniques to pare down the number of possibilities to just 1 trillion.

Then 800 processors at the University of Texas ran for two days to crunch its way through to an answer. After completing its work and spitting out a 200 TB file, the proof revealed that yes, it was possible to color the integers in multiple ways but only up to 7,824. After this point, it was not possible anymore.

The trio verified their proof through another supercomputer.

Although the team has found an answer to the puzzling proof, it did give birth to more questions: why is there a cut-off point at 7,825? Why is the first stretch possible?

But in the meantime, here is another interesting question: did the trio of scientists receive an award for determining the proof?

Yes, they did. Ronald Graham, the mathematician who proposed the Boolean Pythagorean Triples problem in the 1980s, had posed a challenge that whoever solved the problem would get a \$100 reward. Graham made good on his promise and handed over the prize money to the research team.

The team's findings are featured in the Cornell University online library.

Photo: David Goehring | Flickr

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