arXiv:1510.00735math.NT
The 1729 K3 Surface
Early discovery. Linked by one identifiable commenter. This paper is outside the main ranking; a mention is not an endorsement.
Abstract
We revisit the mathematics that Ramanujan developed in connection with the famous "taxi-cab" number $1729$. A study of his writings reveals that he had been studying Euler's diophantine equation $$ a^3+b^3=c^3+d^3. $$ It turns out that Ramanujan's work anticipated deep structures and phenomena which have become fundamental objects in arithmetic geometry and number theory. We find that he discovered a $K3$ surface with Picard number $18$, one which can be used to obtain infinitely many cubic twists over $\mathbb{Q}$ with rank $\geq 2$.
a rather interesting aspect of the taxicab number story is that ramanujan had been looking at "near misses" to the fermat equation, and this was the first values of one of the infinite families of near misses he had constructed. The story is often dramatized to make it sound…
number-sequenceView on HNParent context