In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Ramanujan's ternary quadratic form is the algebraic expression with
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
values for ''x'', ''y'' and ''z''.
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar
(22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
considered this expression in a footnote in a paper
published in 1916 and briefly discussed the representability of integers in this form. After giving
necessary and sufficient conditions
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
that an integer cannot be represented in the form 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 whatever are the values of ''a'', ''b'' and ''c''. It appears, however, that in most cases there are no such simple results."
[ 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[ Ramanujan made the following observations about the form .
*The even numbers that are not of the form are 4λ(16''μ'' + 6).
*The odd numbers that are not of the form , viz. 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[ discovered that the number 679 could not be expressed in the form and they also verified that there were no other such numbers below 2000. This led to an early ]conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
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 . However, in 1941, H Gupta showed that the number 2719 could not be represented as . 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 . Galway verified that there are only eighteen numbers less than 2 × 1010 not representable in the form .[ Based on Galway's computations, Ken Ono and K. Soundararajan formulated the following conjecture:][
:The odd positive integers which are not of the form ''x''2 + are: .
]
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.[
*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 {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
...
then it can be represented in the form .
*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.
References
{{reflist
Srinivasa Ramanujan
Quadratic forms
Squares in number theory
Diophantine equations