1729 — THE TAXICAB NUMBER

G. H. Hardy visited Srinivasa Ramanujan in hospital and, having nothing to say, remarked that he had arrived in taxicab number 1729, which seemed a rather dull number. Ramanujan replied immediately that it was not dull at all: it is the smallest number expressible as the sum of two positive cubes in two different ways. Hardy published the exchange, and it became the most quoted anecdote in mathematics.

The two decompositions are 1 + 1728 and 729 + 1000. Both are exact, and no smaller positive integer admits two such representations — which is what makes the reply a theorem rather than a coincidence noticed on the spot. The general quantity, the smallest number expressible as a sum of two positive cubes in n distinct ways, is now called the n-th taxicab number in memory of the story.

The sequence grows brutally. Ta(1) is 2, Ta(2) is 1729, Ta(3) is 87,539,319, Ta(4) is 6,963,472,309,248, and only six terms are known with certainty. What sounds like a party trick sits on a genuinely hard corner of number theory.

This entry is RESERVED. The number is a strong social signal — it is recognised instantly by people who know it and is invisible to everyone else — but the print has not been designed, and nothing goes into production on the strength of the idea alone.