词条 | Ramanujan's ternary quadratic form |
释义 |
In mathematics, in number theory, Ramanujan's ternary quadratic form is the algebraic expression {{nowrap|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 {{nowrap|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 {{nowrap|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 RamanujanIn his 1916 paper[3] Ramanujan made the following observations about the form {{nowrap|x2 + y2 + 10z2}}.
Odd numbers beyond 391By 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 {{nowrap|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 {{nowrap|x2 + y2 + 10z2}}. However, in 1941, H Gupta[4] showed that the number 2719 could not be represented as {{nowrap|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 programme to determine odd integers not expressible as {{nowrap|x2 + y2 + 10z2}}. Galway verified that there are only eighteen numbers less than {{nowrap|2 × 1010}} not representable in the form {{nowrap|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 + {{nowrap|y2 + 10z2}} are: {{nowrap|3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, 679, 2719}}. Some known resultsThe 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]
References1. ^1 2 3 {{cite journal|first1=Ken|last1=Ono|first2=Kannan | last2=Soundararajan |title=Ramanujan's ternary quadratic form|journal=Inventiones Mathematicae|year=1997|volume=130|issue=3|pages=415–454|url=http://mathcs.emory.edu/~ono/publications-cv/pdfs/025.pdf|doi=10.1007/s002220050191|mr=1483991|citeseerx=10.1.1.585.8840}} 2. ^1 {{cite journal|first1=Burton W. | last1=Jones|first2=Gordon | last2=Pall |title=Regular and semi-regular positive ternary quadratic forms|journal=Acta Mathematica|year=1939|volume=70|issue=1|pages=165–191|doi=10.1007/bf02547347|mr=1555447}} 3. ^1 2 {{cite journal|last=S. Ramanujan|title=On the expression of a number in the form ax2 + by2 + cz2 + du2|journal=Proc. Camb. Phil. Soc.|year=1916|volume=19|pages=11–21}} 4. ^{{cite journal|last=Gupta|first=Hansraj|title=Some idiosyncratic numbers of Ramanujan|journal=Proceedings of the Indian Academy of Sciences, Section A |year=1941|volume=13|issue=6|pages=519–520|url=http://www.ias.ac.in/jarch/proca/13/00000556.pdf|mr=0004816|doi=10.1007/BF03049015}} 5. ^{{cite journal|last=L. E. Dickson|title=Ternary Quadratic Forms and Congruences|journal=Annals of Mathematics |series=Second Series|year=1926–1927|volume=28|issue=1/4|pages=333–341|doi=10.2307/1968378|mr=1502786|jstor=1968378}} 2 : Srinivasa Ramanujan|Quadratic forms |
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