#7825

This number is cool because it appears in two entirely separate areas of math!

First, note that there is something known as the Boolean Pythagorean triples problem. Take the natural numbers 1, 2, ..., n. Color each number either red or blue. Now impose one rule: Every Pythagorean triple must contain both colors.

Recall that a Pythagorean triple is a solution to a² + b² = c² where a, b, and c are integers.

For example, if n = 20, then we can do the following coloring:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Note that there are three Pythagorean triples with all three numbers below 20:

(3, 4, 5)

(5, 12, 13)

(8, 15, 17)

Here, each triple contains both colors. Notice that 5 appears in two triples, creating overlapping constraints. This is what makes the problem interesting. For small n, you can find clever colorings that avoid monochromatic triples. But is it possible to do this forever? Or does it eventually become impossible?

As it turns out, for n ≤ 7,824, such a coloring exists. But as soon as n = 7,825, then a coloring becomes impossible. This is because as n gets big, the Pythagorean triples overlap more and more. They interlock and constrain each other in subtle ways. By the time you reach 7825, the structure becomes so dense that there is no escape.

Unfortunately, the proof is quite ugly. It required reducing the problem to a massive SAT (Boolean satisfiability) instance. A computer then searched all possibilities. The resulting proof file was about 200 terabytes. At the time, it was the largest mathematical proof ever produced. It is far too large for a human to read line by line. Instead, we verify it with proof-checking software.

But hopefully the second fact is more fun :3

A magic square is a square grid of numbers such that every row has the same sum, every column has the same sum, and the two main diagonals also have the same sum.

If the square uses the numbers 1 through n² exactly once, it's called a normal magic square.

Here's the classic 3x3 example of a normal magic square:

8 1 6

​3 5 7

​4 9 2​

with each row, column, and major diagonal summing to 15. We call this number the magic constant.

For a normal magic square, the formula for the magic constant is:

n(n² + 1)/2

If n = 3, this formula gives 15. But if n = 25, we get 7,825.

So this number is used in disorder (breaking the Boolean Pythagorean triples problem) but also order (a magic number for a magic square)!