Manjul Bhargava, a maths professor who just won the Fields Medal, once simplified a 200-year-old number theory law with help of an Indian mathematician’s work from 6th century CE and the popular Rubik’s Cube.
According to an interview the Princeton University professor gave to Quanta Magazine, he said German maths wizard Carl Friedrich Gauss showed that if two numbers – each sum of two perfect squares – are multiplied, the result will also be the sum of two perfect squares.
Bhargava, whose grandfather was a Sanskrit professor in Rajasthan, said he had once seen in Sanskrit manuscripts a generalisation of this same law, credited to Brahmagupta – an Indian mathematician in 628 CE.
The generalisation was this: If two numbers are each the sum of a perfect square and a given whole number times a perfect square are multiplied together, the product will again be sum of a perfect square and that whole number times another perfect square, said the report.
So when Bhargava came across Gauss’ famous 18th century composition law on binary quadratic forms, which had him 20 pages to prove, Bhargava looked for a simpler way to describe the same process.
The process was if one multiplies two binary quadratic forms, the law tells which quadratic form will appear.
The eureka moment, as Bhargava told the magazine, came via Rubik’s Cubes and Mini-Cube (with four squares on each side).
According to the magazine report, Bhargava saw that if he placed numbers on each four corners of the mini-cube and cut the cube in half, the eight corner numbers could be combined to produce a binary quadratic form.
In fact, the cube could generate three binary quadratic forms since there were three ways to cut a cube in half – making front-back, left-right or top-bottom divisions, said the report.
These three forms added up to zero with respect to Gauss’ law. So the cube-slicing method gave a new reformulation of Gauss’ law. Bhargava also found out that if he arranged numbers on a Rubik’s Domino, he could produce a composition law for cubic forms, ones whose exponents are three.
Eventually, he have found 12 more such compositions that became part of his PhD thesis and later, a Benedict Gross, a Harvard mathematician, said the thesis was “first major contribution to Gauss’ theory of composition of binary forms for 200 years”.
When he was awarded the Infosys Prize two years ago, this feat was also cited: “Among Prof. Bhargava’s contributions is the answer to a problem that had eluded the legendary Carl Friedrich Gauss (1777-1855).
“One of Gauss’ discoveries was a law of composition on binary quadratic forms, i.e. expressions of the type ax2+ bxy + cy2, with a, b and c being whole numbers that are fixed, and x and y being the variables. It was an open question as to whether this was isolated or part of a bigger theory.
“Prof. Bhargava showed that quadratic forms were not the only forms with such composition, but that other forms such as cubic forms also have such composition. He was also able to show that the Gauss composition is in fact only one of at least 14 such laws”.