How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

One of the first theorems ever proved in number theory is that of Euclid, which tells us that the primes are infinite in number; we will never run out, no matter how far along the number line we let our minds range. But mathematicians are greedy types, not inclined to be satisfied with a mere assertion of infinitude. After all, there’s infinite and then there’s infinite. There are infinitely many powers of 2, but they’re very rare. Among the first one thousand numbers, there are only ten of them: 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512. There are infinitely many even numbers, too, but they’re much more common: exactly 500 out of the first 1,000 numbers. In fact, it’s pretty apparent that out of the first N numbers, just about (1/2)N will be even. Primes, it turns out, are intermediate—more common than the powers of 2 but rarer than even numbers. Among the first N numbers, about N/log N are prime; this is the Prime Number Theorem, proven at the end of the nineteenth century by the number theorists Jacques Hadamard and Charles-Jean de la Vallée Poussin.