# Srinivasa Ramanujan and Taxicab Numbers

Srinivasa Ramanujan (1887 – 1920) was an Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. He was born in Erode, a small town in Tamil Nadu state, India in a poor Tamil Brahmin family. He was a self-taught mathematician with an uncanny ability to manipulate formulae. Suffering from poverty and moving from one petty job to another as he pursued his mathematical inspirations. After contacting few British mathematicians and not receining a favorable reply, Ramanujan wrote to G H Hardy in 1913. Hardy recognised some of Ramanujan’s formulae but others “seemed scarcely possible to believe”. After he saw Ramanujan’s theorems on continued fractions, Hardy commented that the “[theorems] defeated me completely; I had never seen anything in the least like them before”. He figured that Ramanujan’s theorems “must be true, because, if they were not true, no one would have the imagination to invent them”. Hardy asked a colleague, J E Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were “certainly the most remarkable I have received” and commented that Ramanujan was “a mathematician of the highest quality, a man of altogether exceptional originality and power”. Both Hardy and Littlewood were comparing him with Jacobi, the great German master of formulae. Hardy arranged for him to come to Cambridge, persuading Madras University to grant him a research scholarship. Hardy had a very high opinion of Ramanujan, at least in terms of pure “natural” talent. Ramanujan produced, with Hardy, some remarkable mathematics, and left behind him his extraordinary notebooks, which have since been transcribed and published and continue to be studied, but the English climate and food did not agree with him. He con-tracted tuberculosis, and returned to India in 1918, to die two years later.When Ramanujan was dying of tuberculosis in a hospital, G H Hardy would frequently visit him. It was on one of these visits that the following occurred according to C. P. Snow.

Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two [positive] cubes in two different ways.’

The two different ways are these:

$1729=1^3+12^3=9^3+10^3$

The Hardy-Ramanujan numbers or taxicab numbers are the smallest positive integers that are the sum of 2 cubes of positive integers in $n$ ways. The taxicab numbers from $n=1$ to $n=5$ are as under:

$2=1^3+1^3$

$1729=12^3+1^3=10^3+9^3$

$87539319=228^3+423^3=167^3+436^3=255^3+414^3$

$6963472309248=13322^3+16630^3=10200^3+18072^3=5436^3+18948^3=2421^3+19083^3$

$48988659276962496=231518^3+331954^3=221424^3+336588^3=205292^3+342952^3=107839^3+362753^3=38787^3+365757^3$