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%I A003273 M3747
%S A003273 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 A003273 53,54,55,56,60,61,62,63,65,69,70,71,77,78,79,80,84,85,86,87,88,92,93,
%U A003273 94,95,96,101,102,103,109,110,111,112,116,117,118,119,120,124,125,126
%N A003273 Congruent numbers: positive integers n for which there exists a right 
               triangle having area n and rational sides.
%C A003273 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 A003273 Tunnell shows that if a number is square-free 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 square-free 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 A003273 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 A003273 The Mathematica program for this sequence uses the list of primitive 
               congruent numbers produced by the Mathematica program in A006991.
%C A003273 Conjectured asymptotics (based on random matrix theory) on p. 453 of 
               Cohen's book. [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 23 
               2009]
%D A003273 R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 
               43-45.
%D A003273 R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 
               (1974), 303-305 and 30 (1976), 198.
%D A003273 H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 
               2007, p. 454. [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 23 
               2009]
%D A003273 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 A003273 L. E. Dickson, History of the Theory of Numbers, Vol. 2, pp. 459-472, 
               AMS Chelsea Pub. Providence RI 1999.
%D A003273 R. K. Guy, Unsolved Problems in Number Theory, D27.
%D A003273 G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 
               (1986), 337-340.
%D A003273 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003273 J. B. Tunnell, A classical Diophantine problem and modular forms of weight 
               3/2, Invent. Math., 72 (1983), 323-334.
%H A003273 T. D. Noe, <a href="b003273.txt">Congruent numbers up to 10000; table 
               of n, a(n) for n = 1..5742</a>
%H A003273 American Institute of Mathematics, <a href="http://www.aimath.org/news/
               congruentnumbers/">A trillion triangles</a>
%H A003273 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 A003273 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 A003273 Clay Mathematics Institute, <a href="http://www.claymath.org/prizeproblems/
               birchsd.htm">The Birch and Swinnerton-Dyer Conjecture</a>
%H A003273 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 A003273 E. V. Eikenberg, <a href="http://www.math.umd.edu/~eve/cong_num.html">
               The Congruent Number Problem</a>
%H A003273 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 A003273 Karl Rubin, <a href="http://math.Stanford.EDU/~rubin/lectures/sumo/">
               Elliptic curves and right triangles</a>
%H A003273 W. A. Stein, <a href="http://modular.fas.harvard.edu/edu/Fall2001/124/
               lectures/lecture33/html">Introduction to the Congruent Number Problem</
               a>
%H A003273 W. A. Stein, <a href="http://modular.math.washington.edu/simuw06">The 
               Congruent Number Problem</a>
%H A003273 D. J. Wright, <a href="http://www.math.okstate.edu/~wrightd/4713/nt_essay/
               node7.html">The Congruent Number Problem</a>
%H A003273 Bill Hart, <a href="http://aimath.org/news/congruentnumbers/">A Trillion 
               Triangles</a> [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), 
               Oct 01 2009]
%H A003273 Author?, <a href="http://mathdl.maa.org/mathDL/?pa=mathNews&sa=view&newsId=680">
               Five Mathematicians Capture Record Number of Congruent Numbers</a> 
               [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Oct 01 2009]
%e A003273 24 is congruent because 24 is the area of the right triangle with sides 
               6,8,10.
%t A003273 The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer 
               conjecture:
%t A003273 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 A003273 Cf. A057102, A006991, A072068, A072069, A072070, A072071.
%Y A003273 Sequence in context: A011761 A106745 A165776 this_sequence A006991 A047574 
               A067531
%Y A003273 Adjacent sequences: A003270 A003271 A003272 this_sequence A003274 A003275 
               A003276
%K A003273 nonn,nice
%O A003273 1,1
%A A003273 N. J. A. Sloane (njas(AT)research.att.com).
%E A003273 Guy gives a table up to 1000.
%E A003273 Edited by T. D. Noe (noe(AT)sspectra.com), Jun 14 2002
%E A003273 Comments revised by Max Alekseyev (maxale(AT)gmail.com), Nov 15 2008

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