# Ramanujan's ternary quadratic form

In mathematics, in number theory, **Ramanujan's ternary quadratic form** is the algebraic expression *x*^{2} + *y*^{2} + 10*z*^{2} 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 *ax*^{2} + *by*^{2} + *cz*^{2} 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 *ax*^{2} + *by*^{2} + *cz*^{2} 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 *x*^{2} + *y*^{2} + 10*z*^{2}.

- The even numbers that are not of the form
*x*^{2}+*y*^{2}+ 10*z*^{2}are 4^{λ}(16*μ*+ 6). - The odd numbers that are not of the form
*x*^{2}+*y*^{2}+ 10*z*^{2}, 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 *x*^{2} + *y*^{2} + 10*z*^{2}, 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 *x*^{2} + *y*^{2} + 10*z*^{2} 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 *x*^{2} + *y*^{2} + 10*z*^{2}. However, in 1941, H Gupta^{[4]} showed that the number 2719 could not be represented as *x*^{2} + *y*^{2} + 10*z*^{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 *x*^{2} + *y*^{2} + 10*z*^{2}. Galway verified that there are only eighteen numbers less than 2 × 10^{10} not representable in the form *x*^{2} + *y*^{2} + 10*z*^{2}.^{[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
*x*^{2}+*y*^{2}+ 10*z*^{2}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 10
*n*+ 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*x*^{2}+*y*^{2}+ 10*z*^{2}. - There are only a finite number of odd integers which cannot be represented in the form
*x*^{2}+*y*^{2}+ 10*z*^{2}. - 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

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Ono, Ken; Soundararajan, Kannan (1997). "Ramanujan's ternary quadratic form".*Inventiones Mathematicae***130**(3): 415–454. doi:10.1007/s002220050191. http://mathcs.emory.edu/~ono/publications-cv/pdfs/025.pdf. - ↑
^{2.0}^{2.1}Jones, Burton W.; Pall, Gordon (1939). "Regular and semi-regular positive ternary quadratic forms".*Acta Mathematica***70**(1): 165–191. doi:10.1007/bf02547347. - ↑
^{3.0}^{3.1}^{3.2}S. Ramanujan (1916). "On the expression of a number in the form*ax*^{2}+*by*^{2}+*cz*^{2}+*du*^{2}".*Proc. Camb. Phil. Soc.***19**: 11–21. - ↑ Gupta, Hansraj (1941). "Some idiosyncratic numbers of Ramanujan".
*Proceedings of the Indian Academy of Sciences, Section A***13**(6): 519–520. doi:10.1007/BF03049015. http://www.ias.ac.in/jarch/proca/13/00000556.pdf. - ↑ L. E. Dickson (1926–1927). "Ternary Quadratic Forms and Congruences".
*Annals of Mathematics*. Second Series**28**(1/4): 333–341. doi:10.2307/1968378.

Original source: https://en.wikipedia.org/wiki/ Ramanujan's ternary quadratic form.
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