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A003273 Congruent numbers: positive integers n for which there exists a right triangle having area n and rational sides.
(Formerly M3747)
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126 (list; graph; refs; listen; history; text; internal format)



Positive integers n such that x^2 + n*y^2 = z^2 and x^2 - n*y^2 = t^2 have simultaneous integer solutions. In other words, n is the difference of an arithmetic progression of three rational squares: (t/y)^2, (x/y)^2, (z/y)^2. Values of n 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 n is congruent requires counting the solutions to a pair of equations. For odd n, see A072068 and A072069; for even n see A072070 and A072071.

If a number n is congruent, there are an infinite number of right triangles having rational sides and area n. 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. [From Steven Finch, Apr 23 2009]


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).


T. D. Noe, Congruent numbers up to 10000; table of n, a(n) for n = 1..5742

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

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

A. Dujella, A. S.Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8

E. V. Eikenberg, The Congruent Number Problem

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

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]

MathDL, Five Mathematicians Capture Record Number of Congruent Numbers

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


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


(* 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


Cf. A006991, A072068, A072069, A072070, A072071, A073120, A165564, A182429, A256418, A259680-A259687.

Sequence in context: A011761 A106745 A165776 * A006991 A047574 A273929

Adjacent sequences:  A003270 A003271 A003272 * A003274 A003275 A003276




N. J. A. Sloane


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



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Last modified January 23 09:36 EST 2017. Contains 281185 sequences.