What is the smallest prime number greater than 100?

101, which is also a palindrome and a twin prime with 103. Primes thin out as numbers grow - the density near n falls roughly as one over the natural logarithm of n - but they never stop, which Euclid proved around 300 BC in about three lines.

The proof is the model of elegance. Suppose the primes are a finite list; multiply them all together and add one. The result is not divisible by any prime on the list, since each leaves a remainder of one - so either it is prime itself or has a prime factor missing from the list, and either way the list was incomplete. Nothing has improved on it in 2,300 years.

Primes then went from useless to essential. G.H. Hardy wrote in 1940 that number theory had no practical application and celebrated the fact - and RSA encryption, published in 1977, is built on the difficulty of factoring a large number into two primes. Almost every secure transaction anyone makes relies on it, which makes Hardy's boast one of the more thoroughly overturned predictions in mathematics.