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.
In his 1916 paper [3] Ramanujan made the following observations about the form x2 + y2 + 10z2.
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 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]
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x4 + 4.
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2000 is a natural number following 1999 and preceding 2001.
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In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x2. The adjective which corresponds to squaring is quadratic.
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Ken Ono is an American mathematician with fields of study in number theory. He is the STEM Advisor to the Provost and the Marvin Rosenblum Professor of Mathematics at the University of Virginia.
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