RAMANUJAN, TAXICABS, BIRTHDATES, ZIPCODES, AND TWISTS

It is well known that G. H. Hardy travelled in a taxicab numbered 1729 to an English nursing home to visit his bedridden colleague S. Ramanujan. Hardy was disappointed that his cab had such a mundane number, but to his surprise when he mentioned this to Ramanujan, the brilliant Indian mathematician found 1729 to be quite interesting, for it is the smallest integer that has two distinct representations as a sum of two cubes: 1729 = 13 + 123 = 93 + 103. J. H. Silverman used this famous anecdote to motivate the study of elliptic curves in a recent article [8]. Recently I learned that other permutations of the digits 1, 2, 7, and 9 are significant to the Ramanujan story. Two permutations involve Bruce Berndt, the diligent editor of Ramanujan’s notebooks. Bruce has devoted most of his professional career to undertaking the daunting task of proving many of Ramanujan’s identities (written in notebooks without proofs), but to my surprise his fascination with Ramanujan has profoundly impacted his life outside mathematics. Sonja, Bruce’s youngest daughter, was born in 1972. Is this a coincidence, or could it be an example of “Ramanujan family planning? ” With more sleuthing I discovered that Bruce’s home is in Urbana, Illinois 61802-7219. Could there be any truth to the rumor that Bruce paid the postmaster a mere $12.79 for this vanity zipcode? In a more serious direction, consider the number 2719, which came to my attention in joint work with K. Soundararajan [5]. We begin with the following footnote from Ramanujan’s 1916 paper on quadratic forms [6, p. 14]: “... the even numbers which are not of the form x2 + y2 + 10z2 are the number