To many mathematicians, the mere mention of the number 1729 recalls the following incident involving mathematicians G.H. Hardy and Srinivasa Ramanujan:
Once, in the taxi from London [to Putney], Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen.
"No, Hardy," said Ramanujan. "It is a very interesting number. It is the smallest number expressible as the sum of two [positive] cubes in two different ways." [1]
In memory of this incident, the least number which is the sum of two positive cubes in n different ways is called the nth taxicab number, which I will denote Taxicab(n). In [2], it is shown that for any n >= 1, there indeed exist numbers which are the sum of two positive cubes in n ways, which guarantees the existence of Taxicab(n) for n >= 1.
is so trivial as not to count as a discovery.
was first published by Bernard Frénicle de Bessy in 1657.
was found by Leech in 1957.
was found by E. Rosenstiel, J.A. Dardis, and C.R. Rosenstiel in 1991 [3].
was found by David Wilson on November 21, 1997.
| [1] | R. Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, Washington Square Press, NY, 1991, p. 312. |
| [2] | G.H. Hardy & E.M.Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford University Press, London & NY, 1954, Thm. 412. |
| [3] | E. Rosenstiel, J.A. Dardis & C.R. Rosenstiel, The four least solutions in distinct positive integers of the Diophantine equation s = x3+y3 = z3+w3 = u3+v3 = m3+n3, Bull. Inst. Math. Appl., 27 (1991) pp. 155-157; MR 92i:11134. |