The crisp new dollar bill that I have just taken from my wallet bears the serial number 24598176. It’s easy to tell that the number is exactly divisible by 2 but not by 5. Is it divisible by 3? by 4?

Three and five are prime numbers — that is, they are divisible only by one and by themselves. So are five and seven. And 11 and 13. And for each of these pairs of prime numbers, the difference is two.

In the realm of mathematics, it’s hard to imagine anything more basic than the counting numbers: 1, 2, 3, and so on. Yet this set of mathematical objects abounds with beautiful and unexpected patterns ...

The number 1 is not considered a prime number because it has only 1 positive divisor. Some prime numbers form “twin primes,” like 11 and 13 or 17 and 19. These pairs differ by exactly two. Large prime ...