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%I M3747
%S 5,6,7,13,14,15,20,21,22,23,24,28,29,30,31,34,37,38,39,41,45,46,47,52,
%T 53,54,55,56,60,61,62,63,65,69,70,71,77,78,79,80,84,85,86,87,88,92,93,
%U 94,95,96,101,102,103,109,110,111,112,116,117,118,119,120,124,125,126
%N Congruent numbers: positive integers n for which there exists a right triangle having area n and rational sides.
%C 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 A057102.
%C 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.
%C 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 square number A000290.
%C Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. [From _Steven Finch_, Apr 23 2009]
%D R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 43-45.
%D R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.
%D 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)
%D H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 454. [From _Steven Finch_, Apr 23 2009]
%D R. Cuculiere, "Mille ans de chasse aux nombres congruents", in Pour la Science (French edition of 'Scientific American'), No. 7, 1987, pp. 14-18.
%D L. E. Dickson, History of the Theory of Numbers, Vol. 2, pp. 459-472, AMS Chelsea Pub. Providence RI 1999.
%D R. K. Guy, Unsolved Problems in Number Theory, D27.
%D T. Komatsu, Congruent numbers and continued fractions, Fib. Quart., 50 (2012), 222-226. - From _N. J. A. Sloane_, Mar 04 2013
%D G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
%H T. D. Noe, <a href="/A003273/b003273.txt">Congruent numbers up to 10000; table of n, a(n) for n = 1..5742</a>
%H A. Alvarado and E. H. Goins, <a href="http://arxiv.org/abs/1210.6612">Arithmetic progressions on conic sections</a>, arXiv 2012 [From _Jonathan Sondow_, Oct 25 2012]
%H American Institute of Mathematics, <a href="http://www.aimath.org/news/congruentnumbers/">A trillion triangles</a>
%H E. Brown, Three Fermat Trails to Elliptic Curves, <a href="http://www.math.vt.edu/people/brown/doc/ellip.pdf">5. Congruent Numbers and Elliptic Curves (pp 8-11/17)</a>
%H B. Cipra, <a href="http://sciencenow.sciencemag.org/cgi/content/full/2009/923/3?etoc">Tallying the class of congruent numbers</a>, ScienceNOW, Sep 23 2009
%H Clay Mathematics Institute, <a href="http://www.claymath.org/prizeproblems/birchsd.htm">The Birch and Swinnerton-Dyer Conjecture</a>
%H Department of Pure Maths., Univ. Sheffield, <a href="http://www.shef.ac.uk/~puremath/theorems/congruent.html">Pythagorean triples and the congruent number problem</a>
%H E. V. Eikenberg, <a href="http://www.math.umd.edu/~eve/cong_num.html">The Congruent Number Problem</a>
%H W. F. Hammond, <a href="http://math.albany.edu:8000/math/pers/hammond/Presen/rsumo.html">A Reading of Karl Rubin's SUMO Slides on Rational Right Triangles and Elliptic Curves</a>
%H Bill Hart, <a href="http://aimath.org/news/congruentnumbers/">A Trillion Triangles</a>
%H MathDL, <a href="http://mathdl.maa.org/mathDL/?pa=mathNews&sa=view&newsId=680">Five Mathematicians Capture Record Number of Congruent Numbers</a>
%H Karl Rubin, <a href="http://math.Stanford.EDU/~rubin/lectures/sumo/">Elliptic curves and right triangles</a>
%H W. A. Stein, <a href="http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture33/html">Introduction to the Congruent Number Problem</a>
%H W. A. Stein, <a href="http://modular.math.washington.edu/simuw06">The Congruent Number Problem</a>
%H Ye Tian, <a href="http://arxiv.org/abs/1210.8231">Congruent Numbers and Heegner Points</a>
%H D. J. Wright, <a href="http://www.math.okstate.edu/~wrightd/4713/nt_essay/node7.html">The Congruent Number Problem</a>
%e 24 is congruent because 24 is the area of the right triangle with sides 6,8,10.
%t 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:
%t 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
%Y Cf. A057102, A006991, A072068, A072069, A072070, A072071, A165564, A182429.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_.
%E Guy gives a table up to 1000.
%E Edited by _T. D. Noe_, Jun 14 2002
%E Comments revised by _Max Alekseyev_, Nov 15 2008
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