Taxicab Numbers

Every positive integer was one of his personal friends.

— G.H. Hardy

In 1913, a mathematical prodigy from India, named Srinivasa Ramanujan, started to mail formulae and conjectures to British mathematicians.

Ramanujan had largely self-taught himself numbers, as he surpassed all possible educational opportunities at a young age. By 11, two university students who lived with his family realized he had gleaned everything they knew about math. At 13, he started to develop his own theorems after devouring an advanced book on trigonometry. When he was 15, he learned how to solve cubic equations and then conceived his own method for cracking quartics. Ramanujan received a scholarship to study at the Government Arts College but neglected every other subject to the point of failure. By his mid-20s, he was, in retrospect, clearly a mathematical genius with no training, no degree, and no prospects for future study.

So, he sent out a slew of novel ideas to mathematicians outside of India. In England, the professors who received dossiers dismissed them for their non-rigor. Mathematics had made the full transition to stringent methods and documentation. Proofs ruled and, lacking formal training, Ramanujan presented raw talent and insight instead of the typical finished product.

G.H. Hardy had a different reaction. Hardy received nine pages out of the blue from India. Looking at them, his instincts turned to fraud. He recognized some of the work, but others “seemed scarcely possible to believe.” Thinking about it a little more, he realized the chances of a phony trick seemed remote. He wrote, the theorems “defeated me completely; I had never seen anything in the least like them before,” adding, they “must be true, because, if they were not true, no one would have the imagination to invent them.” Instead of tossing the paper aside, he urged a colleague to view them, who promptly concurred with Hardy. On the basis of nine pages of paper, Hardy deemed Ramanujan “a mathematician of the highest quality, a man of altogether exceptional originality and power.” Yet, another mathematician wrote “not one [theorem] could have been set in the most advanced mathematical examination in the world.”
Srinivasa Ramanujan - photo by Konrad Jacobs
G.H. Hardy - unknown photographer

Hardy would know the real deal when he encountered it. He penned A course of pure mathematics in 1908, a text that is still in print in the 21st century. He formulated a law that is the foundation for population genetics that now bears his name: the Hardy–Weinberg law. Hardy contributed greatly to the field of number theory.

Later in life, despite all his achievements, Hardy recounted that his greatest contribution to the field was the “discovery” of Ramanujan.

Hardy worked to bring Ramanujan to England, hoping to mold a massive innate talent into a formal mathematical powerhouse. When he finally arrived, the two methods of mathematics – formulation and intuition – clashed. Hardy wanted to mentor formality while maintaining creativity. Professors found it difficult to instruct Ramanujan in proofs or rigor. One step forward in education would yield to multiple original ideas that Ramanujan pursued with his own methods.

Still, Ramanujan remained in England for five years, studying with Hardy and others. The period generated some of the most original mathematical work in history. Hardy claimed to have never met his equal, comparing him to some of the great ghosts of the field, Euler and Jacobi.

Ramanujan at center; Hardy rightmost; Cambridge circa 1915

Why is Srinivasa Ramanujan not a more famous name to those outside of mathematics? He generated more than 3,900 identities and equations, many of which now officially assume his name, such as the Ramanujan prime or the Ramanujan theta function. Though unconventionally offered, most of his work was later proved by formal mathematicians. His notebooks continue to provide inspiration for new work.

Being from India in an era dominated by Europe certainly didn’t help, nor did his reluctance to adhere to traditions. Perhaps the biggest factor, however, was the fact that Ramanujan died at age 32. His proliferation during such a short lifespan defies belief. He had suffered twice from dysentery earlier in life and, in 1920, succumbed to hepatic amoebiasis, a result of his earlier illnesses.

Before death, Hardy visited his friend in the hospital, where their encounter produced the most lasting image related to Ramanujan.

A taxi from the film The Man Who Knew Infinity

Hardy wrote:

“I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”

Though he did not invent the notion, Ramanujan had recognized the number immediately. One can compute 1729 by summing the cubes of two different sets of positive integers:

1729 = 13 + 123 = 93 + 103

Today, we call such oddities taxicab numbers, after the famous impetus for further study of the concept. A taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. 1729 is the smallest number in which two cubes can sum in two distinct ways. Put another way, Taxicab(2) = 1729.

Ramanujan did not live to see the fruits, but Hardy and E.M. Wright proved in 1938 that such numbers exist for all positive integers n. Beyond n = 2, things get really large, really quickly because the number of distinct pairs of cubes needs to increase with each step up the ladder. Taxicab(3), for example, has three distinct pairs:

Ta(3) = 87539319 = 1673 + 4363 = 2283 + 4233 = 2553 + 4143

The third number was not found until 1957. In 1991, a group discovered Ta(4) = 6,963,472,309,248, with four pairs of cubes summing to that monster number. The fifth in the series fell in 1999, followed by the sixth in 2003 (48988659276962496 and 24153319581254312065344). Things get so out of hand after this point that researchers only know the upper bounds of subsequent taxicab numbers.

Two uses of the number in Futurama

Taxicab(2) – 1729 – is known as the Hardy-Ramanujan Number.

What could Srinivasa Ramanujan have accomplished if he survived past his early 30s? Famous mathematician Paul Erdos relayed Hardy’s firsthand ratings of the intellects he encountered. On a scale of 0 to 100, he gave himself 25, his colleague and fellow admirer of Ramanujan, J.E. Littlewood, a 30, David Hilbert, one of the most influential mathematicians of the 19th and 20th centuries, an 80, while reserving a perfect 100 for Ramanujan. When Ramanujan’s “Lost Notebook” was discovered in 1976, containing discoveries from the final years of his life, a seismic wave passed through the mathematical world, sending number lovers poring over the treasure trove. Though he never became a rigorous prover, his equations have time and again been proven correct.

Numbers might not hold much significance for many people, but, for others, they are artistic and creative entities. The number 1729 might seem to be a random string of four digits, yet Srinivasa Ramanujan recognized a beauty. Thanks to a random taxi, we have a splendid insight into his love of numbers.

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