A Massive Amount of Data Just to Solve a Problem. Is This Problem Really That Difficult?
Many of us have spent years studying mathematics from elementary to high school just to escape from it, only to encounter Advanced Mathematics in college again. Do you think the math homework assigned after each class is tough? Well, take a look at this problem: it took three mathematicians and 200 terabytes of data just to store the solution, aided by a supercomputer!
To put this into perspective, one terabyte can hold 337,920 copies of War and Peace, the longest novel in human history by Leo Tolstoy. So, imagine how much text 200 terabytes can contain.
How difficult is this problem that it requires such a monumental solution? It’s a mathematical issue revolving around the Pythagorean Theorem, introduced by mathematician Ronald Graham in the 1980s. The problem is called Boolean Pythagorean Triples and was so challenging that Graham offered a $100 reward to anyone who could solve it (back in 1980!).
The mathematical issue relates to the formula of the Pythagorean Theorem: a² + b² = c², where a and b are the two legs of a right triangle, and c is the hypotenuse.
The formula of the Pythagorean Theorem.
Explaining the name of this mathematical problem:
A Boolean variable can have a value of either true or false.
As for Pythagorean Triples, certain sets of positive integers known as Pythagorean triples always satisfy the Pythagorean theorem, such as: 3² + 4² = 5²; 8² + 15² = 17². These are referred to as Pythagorean Triples.
Now imagine that every positive integer in the number line is colored either red or blue. Graham posed the question: is it possible to color every integer either blue or red such that no Pythagorean triple shares the same color? A reward of $100 would be given to anyone who could solve this problem (With $100, you could buy a one-terabyte storage drive!).
Ronald Graham, the teacher who posed a problem that took decades to solve.
This mathematical problem is difficult because: a positive integer can belong to multiple Pythagorean Triples. For example, the number 5 belongs to the triple 3-4-5, but it also belongs to the triple 5-12-13. According to Graham’s condition, if the number 5 in the first triple is colored blue, then in the second triple, it must also be blue, which means the numbers 12 and 13 must be red.
As we dig deeper into Graham’s conditions, the numbers get larger, and the problem begins to arise. If 12 must be red in the triple 5-12-13, then any subsequent triples containing the number 12 must follow a specific coloring.
Mathematicians Marijn Heule from the University of Texas, Victor Marek from the University of Kentucky, and Oliver Kullmann from Swansea University in the UK collaborated to tackle this problem. They implemented several tests and computational techniques on the Stampede supercomputer at the University of Texas, allowing them to narrow down the “coloring” possibilities to 102,300 trillion (that’s a total of 25 zeros!).
The Stampede supercomputer used to solve this difficult problem.
This supercomputer, equipped with 800 powerful processors, took two days to “process” all those tests, and it was only feasible for numbers up to 7,824. Beyond that, starting from 7,825, it was impossible to meet Graham’s conditions.
Thus, three mathematicians (along with a supercomputer) solved a mathematical problem that had persisted for decades, and Ronald Graham kept his promise by generously awarding the $100 prize to the three of them.
“The atomic trio” of these mathematicians generated a compressed file of 68 gigabytes for any young individual with a good processor and 30,000 free hours to download, reconstruct, and verify the issue. However, if you truly had 30,000 free hours, there would be another problem: humans cannot read those lines of code.
In fact, the trio had to “rely on” another computer program to verify their results, and ultimately, 7,824 was the correct number. Ronald Graham was satisfied with the verification of this number.
Numbers from 1 to 7,824 can be colored as they satisfy Graham’s conditions. However, from 7,825 onwards, no number satisfies the conditions.
However, many believe that because humans cannot read the results, it is not convincing enough. Although it cannot be proven wrong, this does not resolve the issue entirely. Why is it impossible to “color” starting from the number 7,825? We cannot explain it; we just have to take the supercomputer’s word for it.
How can we make sense of the meaning of numbers for us and for the Universe if all mathematical problems are solved by machines like this? The reality is that this problem is too difficult to solve; perhaps it will require another supercomputer to intervene.
