In 1737, Leonhard Euler ingeniously proved that the sum of the reciprocals of the prime numbers also diverges. Paul Erdős gave another proof of this result via a particularly elegant proof by contradiction in 1938.
It was Paul Erdős who took the idea a little further, asking what made the set P of prime numbers so special in this context. Let A be a set of natural numbers not containing 1, such that none of its elements divides another. This is indeed true of P; you will easily find other such sets, finite or infinite, with this property. Paul Erdős called these sets primitive and proved in 1935 that the sum is always finite.
Better still, all these sums are bounded above by a certain absolute constant, independent of the choice of primitive set! In 1988, while visiting Limoges (Haute-Vienne), Paul Erdős even conjectured that the largest of these sums was in fact the one obtained from the set P of prime numbers.
