Ramanujan’s Taxicab Numbers are the smallest numbers that can be written as a sum of two numbers to the power of n in two different ways. We only know such numbers if n is 1, 2, 3 and 4. We don’t know the smallest number that can be written as a sum of two fifth powers in two different ways, or any other powers larger than four. The name “Taxicab Numbers” comes from the story about Ramanujan’s visit to Cambridge, when he got picked up by a cab with 1729 on the number plate.
“Every possible integer is one of Ramanujan’s personal friends”, Simon tells me after he sketches the Taxicab numbers at a beach cafe. “Which is a bit uncomfortable because you have infinitely many friends then”.