Ramanujan's ternary quadratic form

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In mathematics, in number theory, Ramanujan's ternary quadratic form is the algebraic expression x2 + y2 + 10z2 with integral values for x, y and z.[1][2] Srinivasa Ramanujan considered this expression in a footnote in a paper[3] published in 1916 and briefly discussed the representability of integers in this form. After giving necessary and sufficient conditions that an integer cannot be represented in the form ax2 + by2 + cz2 for certain specific values of a, b and c, Ramanujan observed in a footnote: "(These) results may tempt us to suppose that there are similar simple results for the form ax2 + by2 + cz2 whatever are the values of a, b and c. It appears, however, that in most cases there are no such simple results."[3] To substantiate this observation, Ramanujan discussed the form which is now referred to as Ramanujan's ternary quadratic form.

Properties discovered by Ramanujan[edit]

In his 1916 paper[3] Ramanujan made the following observations about the form x2 + y2 + 10z2.

  • The even numbers which are not of the form x2 + y2 + 10z2 are 4λ(16μ + 6).
  • The odd numbers that are not of the form x2 + y2 + 10z2, viz. 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, . . . do not seem to obey any simple law.

Odd numbers beyond 391[edit]

By putting an ellipsis at the end of the list of odd numbers not representable as x2 + y2 + 10z2, Ramanujan indicated that his list was incomplete. It was not clear whether Ramanujan intended it to be a finite list or infinite list. This prompted others to look for such odd numbers. In 1927, Burton W. Jones and Gordon Pall[2] discovered that the number 679 could not be expressed in the form x2 + y2 + 10z2 and they also verified that there were no other such numbers below 2000. This prompted them to surmise that the seventeen numbers - the sixteen numbers in Ramanujan's list and the number discovered by them - were the only odd numbers not representable as x2 + y2 + 10z2. However, in 1941, H Gupta[4] showed that the number 2719 could not be represented as x2 + y2 + 10z.2 He also verified that there were no other such numbers below 20000. Further progress in this direction took place only after the development of modern computers. W. Galway wrote a computer programme to determine odd integers not expressible as x2 + y2 + 10z2. Galway verified that there are only eighteen numbers less than 2 × 1010 not representable in the form x2 + y2 + 10z2.[1] Based on Galway's computations, Ken Ono and K. Soundararajan formulated the following conjecture:[1]

The odd positive integers which are not of the form x2 + y2 + 10z2 are: 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, 679, 2719.

Some known results about Ramanujan's ternary quadratic form[edit]

The conjecture of Ken Ono and Soundararajan has not been fully resolved. However, besides the results enunciated by Ramanujan, a few more general results about the form have been established. The proofs of some of them are quite simple while those of the others involve quite complicated concepts and arguments.[1]

  • Every integer of the form 10n + 5 is represented by Ramanujan's ternary quadratic form.
  • If n is an odd integer which is not square-free then it can be represented in the form x2 + y2 + 10z2.
  • There are only a finite number of odd integers which cannot be represented in the form x2 + y2 + 10z2.
  • If the generalized Riemann hypothesis is true, then the conjecture of Ono and Soundararajan is also true.
  • Ramanujan's ternary quadratic form is not regular in the sense of L.E. Dickson.[5]

References[edit]

  1. ^ a b c d Ken Ono; K. Soundararajan (1997). "Ramanujan's ternary quadratic form". Inventiones Mathematicae 130 (3): 415–454. doi:10.1007/s002220050191. Retrieved 30 November 2012. 
  2. ^ a b Burton W. Jones; Gordon Pall (1939). "Regular and semi-regular positive ternary quadratic forms". Acta Mathematica 70 (1): 165–191. doi:10.1007/bf02547347. Retrieved 1 December 2012. 
  3. ^ a b c S. Ramanujan, (1916). "On the expression of a number in the form ax2 + by2 + cz2 + du2". Proc. Camb. Phil. Soc. 19: 11–21. 
  4. ^ Gupta, Hansraj (June 1941). "Some idiosyncratic numbers of Ramanujan". Proceedings of the Indian Academy of Sciences – Section A June 1941, Volume 13, Issue 6, pp 519–520 13 (6): 519–520. Retrieved 12 November 2012. 
  5. ^ L. E. Dickson (1926–1927). "Ternary Quadratic Forms and Congruences". Annals of Mathematics. Second Series 28 (1/4): 333–341. doi:10.2307/1968378. Retrieved 1 December 2012.