Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited

Read  full  paper  at:http://www.scirp.org/journal/PaperInformation.aspx?PaperID=55718#.VS9ER9KqpBc

Author(s)

ABSTRACT

Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x<sup< em=””>2+by<sup< em=””><sup< em=””>2+cz<sup< em=””>2<span “=””> with b,c {1,2,4,8}<span “=””>. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.

Cite this paper

Hürlimann, W. (2015) Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited.Advances in Pure Mathematics, 5, 267-277. doi: 10.4236/apm.2015.55027.

References

[1] Gauss, C.F. (1801) Disquitiones Arithmeticae. Fleischer, Leipzig.
[2] Dickson, L.E. (1920) History of the Theory of Numbers, Vol. II. Carnegie Institute of Washington, Washington.
[3] Cooper, S. and Hirschhorn, M.D. (2007) On the Number of Primitive Representations of Integers as Sums of Squares. Ramanujan Journal, 13, 7-25.
http://dx.doi.org/10.1007/s11139-006-0240-6
[4] Hirschhorn, M.D. and Sellers, J.A. (1999) On Representations of a Number as a Sum of Three Squares. Discrete Mathematics, 199, 85-101.
http://dx.doi.org/10.1016/S0012-365X(98)00288-X
[5] Bell, E.T. (1924) The Numbers of Representations of Integers in Certain Forms ax2 + by2 + cz2. American Mathematical Monthly, 31, 126-131.
http://dx.doi.org/10.2307/2299890
[6] Dickson, L.E. (1923) History of the Theory of Numbers, Vol. III. Carnegie Institute of Washington, Washington.
[7] Tunnel, J. (1983) A Classical Diophantine Problem and Modular Forms of Weight 3/2. Inventiones Mathematicae, 72, 323-334.
http://dx.doi.org/10.1007/BF01389327
[8] Mordell, L.J. (1969) Diophantine Equations. Pure and Applied Mathematics, Vol. 30, London and New York.
[9] Koblitz, N. (1984) Introduction to Elliptic Curves and Modular Forms. Springer, New York.
http://dx.doi.org/10.1007/978-1-4684-0255-1
[10] Hürlimann, W. (2011) A Congruent Twin Number Problem. Pioneer Journal of Algebra, Number Theory and Its Applications, 1, 53-66.
[11] Cohen, H. (2007) Number Theory, Volume I: Tools and Diophantine Equations (Graduate Texts in Mathematics). Springer Science + Business Media, LLC, New York.
[12] Hurwitz, A. (1886) Ueber die Anzahl der Classen Quadratischer Formen von Negativer Diskriminante. Journal für Diereine und Angewandte Mathematik, 99, 165-168.
[13] Cooper, S. and Hirschhorn, M.D. (2004) Results of Hurwitz Type for Three Squares. Discrete Mathematics, 274, 9-24.
http://dx.doi.org/10.1016/S0012-365X(03)00079-7
[14] Berndt, B.C. (1991) Ramanujan’s Notebooks, Part III. Springer, New York.
http://dx.doi.org/10.1007/978-1-4612-0965-2
[15] Bateman, P.T. and Knopp, M.I. (1998) Some New Old-Fashioned Modular Identities. The Ramanujan Journal, 2, 247-269.
http://dx.doi.org/10.1023/A:1009782529605
[16] Barrucand, P., Cooper, S. and Hirschhorn, M.D. (1998) Relations between Squares and Triangles. Discrete Mathematics, 248, 245-247.
http://dx.doi.org/10.1016/S0012-365X(01)00344-2
[17] Cooper, S. and Hirschhorn, M.D. (2004) A Combinatorial Proof of a Result from Number Theory. Integers, 4, Paper A09.
[18] Nagell, T. (1929) L’analyse indéterminée de degré supérieur. Gauthier-Villars, Paris.
[19] Bastien, L. (1915) Nombres Congruents. L’Intermédiaire des Mathématiciens, 22, 231-232.
[20] Heegner, K. (1952) Diophantische Analysis und Modulfunktionen. Mathematische Zeitschrift, 56, 227-253.
http://dx.doi.org/10.1007/BF01174749
[21] Birch, B.J. (1968) Diophantine Analysis and Modular Functions. Oxford University Press, Oxford, 35-42.
[22] Stephens, N.M. (1975) Congruence Properties of Congruent Numbers. Bulletin of the London Mathematical Society, 7, 182-184.
http://dx.doi.org/10.1112/blms/7.2.182
[23] Alter, R., Curtz, T.B. and Kubota, K.K. (1972) Remarks and Results on Congruent Numbers. Proceedings of the 3rd Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, 28 February-2 March 1972, 27-35.
[24] Conrad, K. (2008) The Congruent Number Problem. Harvard College Mathematical Review, 2, 58-73.
[25] Ono, T. (1994) Variations on a Theme of Euler. Quadratic Forms, Elliptic Curves, and Hopf Maps. Plenum Press, New York and London.
http://dx.doi.org/10.1007/978-1-4757-2326-7
Advertisements

发表评论

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / 更改 )

Twitter picture

You are commenting using your Twitter account. Log Out / 更改 )

Facebook photo

You are commenting using your Facebook account. Log Out / 更改 )

Google+ photo

You are commenting using your Google+ account. Log Out / 更改 )

Connecting to %s