OFFSET
1,1
COMMENTS
Positive integers k such that x^2 + k*y^2 = z^2 and x^2 - k*y^2 = t^2 have simultaneous integer solutions. In other words, k is the difference of an arithmetic progression of three rational squares: (t/y)^2, (x/y)^2, (z/y)^2. Values of k corresponding to y=1 (i.e., an arithmetic progression of three integer squares) form A256418.
Tunnell shows that if a number is squarefree and congruent, then the ratio of the number of solutions of a pair of equations is 2. If the Birch and Swinnerton-Dyer conjecture is assumed, then determining whether a squarefree number k is congruent requires counting the solutions to a pair of equations. For odd k, see A072068 and A072069; for even k see A072070 and A072071.
If a number k is congruent, there are an infinite number of right triangles having rational sides and area k. All congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a positive square number A000290.
Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. - Steven Finch, Apr 23 2009
REFERENCES
Alter, Ronald; Curtz, Thaddeus B.; Kubota, K. K. Remarks and results on congruent numbers. Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1972), pp. 27-35. Florida Atlantic Univ., Boca Raton, Fla., 1972. MR0349554 (50 #2047)
H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 454. [From Steven Finch, Apr 23 2009]
R. Cuculière, "Mille ans de chasse aux nombres congruents", in Pour la Science (French edition of 'Scientific American'), No. 7, 1987, pp. 14-18.
L. E. Dickson, History of the Theory of Numbers, Vol. 2, pp. 459-472, AMS Chelsea Pub. Providence RI 1999.
R. K. Guy, Unsolved Problems in Number Theory, D27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
R. Alter, Letter to N. J. A. Sloane, Sep 1975
R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 43-45.
R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.
A. Alvarado and E. H. Goins, Arithmetic progressions on conic sections, arXiv:1210.6612 [math.NT], 2012. [From Jonathan Sondow, Oct 25 2012]
E. Brown, Three Fermat Trails to Elliptic Curves, 5. Congruent Numbers and Elliptic Curves (pp 8-11/17)
Graeme Brown, The Congruent Number Problem, 2014.
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009.
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Raiza Corpuz, Constructing congruent number elliptic curves using 2-descent, arXiv:2006.08113 [math.NT], 2020.
R. Cuculière, Mille ans de chasse aux nombres congruents, Séminaire de Philosophie et Mathématiques, 2, 1988, p. 1-17.
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem [Cached copy]
A. Dujella, A. S.Janfeda, and S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8.
E. V. Eikenberg, The Congruent Number Problem
David Goldberg, Triangle Sides for Congruent Numbers less than 10000, arXiv:2106.07373 [math.NT], 2021.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent number elliptic curves with rank at least two, arXiv:1809.02037 [math.NT], 2018.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent Number Elliptic Curves Related to Integral Solutions of m^2 = n^2 + nl + n^2, J. Int. Seq., Vol. 22 (2019), Article 19.3.1.
Alvaro Lozano-Robledo, My #MegaFavNumber: 224,403,517,704,336,969,924,557,513,090,674,863,160,948,472,041, video (2020) [discusses congruent numbers and a(157)]
W. F. Hammond, A Reading of Karl Rubin's SUMO Slides on Rational Right Triangles and Elliptic Curves
Bill Hart, A Trillion Triangles, American Institute of Mathematics.
T. Komatsu, Congruent numbers and continued fractions, Fib. Quart., 50 (2012), 222-226. - From N. J. A. Sloane, Mar 04 2013
S. Komoto, T. Watanabe and H. Wada, 42553 is a congruent number.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]
Allan J. MacLeod, The congruent number descent of Komotu, Watanabe and Wada, arXiv:2005.02615 [math.NT], 2020.
Fidel Ronquillo Nemenzo, All congruent numbers less than 40000, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31.
Karl Rubin, Right triangles and elliptic curves
W. A. Stein, Introduction to the Congruent Number Problem
W. A. Stein, The Congruent Number Problem
Ye Tian, Congruent Numbers and Heegner Points, arXiv:1210.8231 [math.NT], 2012.
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
D. J. Wright, The Congruent Number Problem
EXAMPLE
24 is congruent because 24 is the area of the right triangle with sides 6,8,10.
5 is congruent because 5 is the area of the right triangle with sides 3/2, 20/3, 41/6 (although not of any right triangle with integer sides -- see A073120). - Jonathan Sondow, Oct 04 2013
MATHEMATICA
(* The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer conjecture and uses the list of primitive congruent numbers produced by the Mathematica code in A006991: *)
For[cLst={}; i=1, i<=Length[lst], i++, n=lst[[i]]; j=1; While[n j^2<=maxN, cLst=Union[cLst, {n j^2}]; j++ ]]; cLst
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Guy gives a table up to 1000.
Edited by T. D. Noe, Jun 14 2002
Comments revised by Max Alekseyev, Nov 15 2008
Comment corrected by Jonathan Sondow, Oct 10 2013
STATUS
approved