Ramanujan's ternary quadratic form

In number theory, a branch of mathematics, 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

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

• The even numbers that are not of the form x² + y² + 10z² are 4^λ(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

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 led to an early conjecture 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 + 10z². 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 program 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

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

1. Ono, Ken; Soundararajan, Kannan (1997). "Ramanujan's ternary quadratic form" (PDF). Inventiones Mathematicae. 130 (3): 415–454. Bibcode:1997InMat.130..415O. CiteSeerX 10.1.1.585.8840. doi:10.1007/s002220050191. MR 1483991. S2CID 122314044.
2. Jones, Burton W.; Pall, Gordon (1939). "Regular and semi-regular positive ternary quadratic forms". Acta Mathematica. 70 (1): 165–191. doi:10.1007/bf02547347. MR 1555447.
3. Ramanujan, S. (1916). "On the expression of a number in the form ax² + by² + cz² + du²". Mathematical Proceedings of the Cambridge Philosophical Society. 19: 11–21.
4. Gupta, Hansraj (1941). "Some idiosyncratic numbers of Ramanujan" (PDF). Proceedings of the Indian Academy of Sciences, Section A. 13 (6): 519–520. doi:10.1007/BF03049015. MR 0004816. S2CID 116006923.
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. JSTOR 1968378. MR 1502786.