From: cet1@cus.cam.ac.uk (Chris Thompson)
Newsgroups: sci.math
Subject: Re: Ramanujan Quintuples - references?
Date: 28 Mar 1995 13:31:31 GMT

In article <3kvkqm$7g3@mcmail.CIS.McMaster.CA>, kovarik@mcmail.cis.mcmaster.ca 
(Zdislav V. Kovarik) writes:
|>    From a story about Ramanujan came a definition: a quintuple (N,a,b,c,d)
|> of positive integers is a Ramanujan quintuple if N = a^3 + b^3 = c^3 + d^3
|> where the unordered pairs (a,b) and (c,d) are different. The original
|> (minimal) example is (1729, 1, 12, 9, 10). 
|>    Is there a book or journal study of such decompositions?

Well, it's a wee bit over-specific, don't you think? On the other hand, 
you can start from here and work your way into the vast fields of
Diophantine analysis and/or elliptic curves. A good place to start
would be to look at some of the references given in section D1 of
Richard Guy's "Unsolved Problems in Number Theory" (2nd edition, Springer,
1994, ISBN 0-387-94289-0).

To whet your appetite, a few facts at random:

  The smallest number expressible as the sum of two positive cubes in
  *three* different ways is 87539319 = 167^3 + 436^3 = 228^3 + 423^3 =
  = 255^3 + 414^3 (Leech, 1957).

  The smallest expessible in *four* different ways is 6963472309248 =
  2421^3 + 19083^3 = 5436^3 + 18948^3 = 10200^3 + 18072^3 = 13322^3 + 16630^3
  (Rosenstiel,Dardis,Rosenstiel, 1991). [UPINT has a typo here: 15530 for 16630]

  There are numbers expressible in arbitrarily many different ways (but
  finding the smallest such isn't easy). See Hardy and Wright [UPINT says
  Theorem 412, but it is 413 in the 2nd edition]. Then work out the sense
  in which the algebraic argument there is equivalent to "the rational
  points on an elliptic curve of non-zero rank are dense on [the unbounded
  components of] the curve over the reals".

  You can make things easier by allowing negative cubes, e.g. 91 = 4^3 + 3^3 =
  = 6^3 + (-5)^3. Or more difficult, by allowing only decompositions into
  coprime cubes. In this case it is *not* known that there are integers
  with arbitrarily many decompositions: in fact, by a result of Joseph
  Silverman, that would imply the existence of elliptic curves of arbitrarily
  large rank.

  Or you can turn your attention to equal sums of fourth powers, and look at
  the long history (since Euler) of parametric (and other) solutions to
  a^4 + b^4 = c^4 + d^4. No solution to a^4 + b^4 = c^4 + d^4 = e^4 + f^4
  is yet known...

  And so on, indefinitely. You are never going to exhaust the subject.

To return to Ramanujan, Hardy, and taxi cabs: This property of 1729 was,
of course, noticed long before the incident in question [UPINT says de Bessy,
1657]. In some ways, the story tells one as much about Hardy (for not noticing 
it) as about Ramanujan. Some mathematicians keep trivia-that-might-illuminate-
deep-truths like this in rapidly accessible memory, and some do not. It
probably doesn't say much about their quality as mathematicians either way.

Chris Thompson
Internet: cet1@cus.cam.ac.uk
JANET:    cet1@uk.ac.cam.cus