SRINIVASA RAMANUJAN

In his short life, Srinivasa Ramanujan made contributions to analysis, number theory, and infinite series. With G.H. Hardy, he discovered the first taxicab number: 1729.

• December 22, 1887 Ramanujan born in Erode, Madras, India. In childhood, he survives smallpox but later suffers bouts of dysentery
• 1892–1907 Ramanujan excels in mathematics at school and college. He has little interest in any other subject and twice fails his Fellow of Arts exam
• 1910 Becomes a researcher at the University of Madras
• 1913 Ramanujan writes to G.H. Hardy, a Cambridge lecturer, presenting some of his theorems. Hardy recognizes genius and invites him to study at Trinity College, under a scholarship. Ramanujan goes on to spend five years at Cambridge
• 1918 Becomes the first Indian person elected as a fellow of Trinity College, Cambridge. He also becomes a member of the Royal Society
• April 26, 1920 After contracting tuberculosis, Ramanujan returns to Madras, where he dies aged thirty-two

TAXICAB NUMBERS

G.H. Hardy visited Ramanujan in a London hospital and traveled in a taxi with the number 1729. Hardy remarked it was “rather a dull number,” but Ramanujan contradicted him, saying “No, it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”

The two ways are:

1729 = 1³ + 12³ = 9³ + 10³

1729 is also known as the “Hardy–Ramanujan number.”

In general, the taxicab numbers, Ta(n), are the smallest numbers that can be expressed as the sum of two cubes in n distinct ways.

n	Ta(n)	sums of cubes
1	2	13 + 13
2	1729	13 + 123
		93 + 103
3	87539319	1673 + 4363
		2283 + 4233
		2553 + 4143
4	6963472309248	24213 + 190833
		54363 + 189483
		102003 + 180723
		133223 + 166303
5	48988659276962496	387873 + 3657573
		1078393 + 3627533
		2052923 + 3429523
		2214243 + 3365883
		2315183 + 3319543
6	24153319581254312065344	5821623 + 289062063
		30641733 + 288948033
		85192813 + 286574873
		162180683 + 270932083
		174924963 + 265904523
		182899223 + 262243663