Upward Spiral

They say you have to hit rock bottom before you start to rise back up. Don't take this as fact, but rock bottom is where I imagine the mathematician Stanislaw Ulam making his most interesting discovery.

"I've got nothing!" he might have lamented, as he scrawled numbers in a spiral to kill time...

"Wait a second... why are all the primes on diagonals?"

Primes, the natural numbers that are not multiples of anything smaller, are an endlessly fascinating set. Any number can be written as the product of primes (for example, 33 = 3 x 11). Thus they are the "atoms" of the math world - the irreducible numbers from which all others are constructed.

Furthermore, a whole gammit of crazy statements seem to be true about them, though many can't be proven: There is at least one prime between any number and twice that number. There are an infinite set of primes, the largest currently known being 4 million digits long. There are even an infinite set of "twin primes" which are separated by 2, like 29 and 31. Another unproven but seemingly true conjecture is that every even number (besides 2) is the sum of two primes, which is quite unrelated to other multiplicative properties.

And like Ulam noticed, primes appear along diagonals when numbers are written in a spiral. In this large block of spiraling numbers, the white dots are primes.

Stranger still is the fact that some diagonals are more strongly prime than others. Why should this be? Unfortunately there's no telling; we really don't understand the distribution of the primes. But if we had to convey our intelligence to an alien life form, many scientists think the best way would be to demonstrate our knowledge of these incredible numbers. As the building blocks of math, they are the true universal language.