# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a003273 Showing 1-1 of 1 %I A003273 M3747 #142 Oct 28 2024 17:09:10 %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 k for which there exists a right triangle having area k and rational sides. %C A003273 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. %C A003273 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. %C A003273 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. %C A003273 Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. - _Steven Finch_, Apr 23 2009 %D A003273 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 A003273 H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 454. [From _Steven Finch_, Apr 23 2009] %D A003273 R. Cuculière, "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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A003273 T. D. Noe, Congruent numbers up to 10000; table of n, a(n) for n = 1..5742 %H A003273 R. Alter, Letter to N. J. A. Sloane, Sep 1975 %H A003273 R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 43-45. %H A003273 R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198. %H A003273 A. Alvarado and E. H. Goins, Arithmetic progressions on conic sections, arXiv:1210.6612 [math.NT], 2012. [From _Jonathan Sondow_, Oct 25 2012] %H A003273 Estelle Basor and Bill Hart, A trillion triangles, American Institute of Mathematics, %H A003273 E. Brown, Three Fermat Trails to Elliptic Curves, 5. Congruent Numbers and Elliptic Curves (pp 8-11/17) %H A003273 Graeme Brown, The Congruent Number Problem, 2014. %H A003273 Jasbir S. Chahal, Some remarks on rational right triangles, Expos. Math. (2024). %H A003273 B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009. %H A003273 Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture %H A003273 Raiza Corpuz, Constructing congruent number elliptic curves using 2-descent, arXiv:2006.08113 [math.NT], 2020. %H A003273 R. Cuculière, Mille ans de chasse aux nombres congruents, Séminaire de Philosophie et Mathématiques, 2, 1988, p. 1-17. %H A003273 Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem [Cached copy] %H A003273 A. Dujella, A. S.Janfeda, and S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8. %H A003273 E. V. Eikenberg, The Congruent Number Problem %H A003273 David Goldberg, Triangle Sides for Congruent Numbers less than 10000, arXiv:2106.07373 [math.NT], 2021. %H A003273 Lorenz Halbeisen and Norbert Hungerbühler, Congruent number elliptic curves with rank at least two, arXiv:1809.02037 [math.NT], 2018. %H A003273 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. %H A003273 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)] %H A003273 W. F. Hammond, A Reading of Karl Rubin's SUMO Slides on Rational Right Triangles and Elliptic Curves %H A003273 Bill Hart, A Trillion Triangles, American Institute of Mathematics. %H A003273 T. Komatsu, Congruent numbers and continued fractions, Fib. Quart., 50 (2012), 222-226. - From _N. J. A. Sloane_, Mar 04 2013 %H A003273 S. Komoto, T. Watanabe and H. Wada, 42553 is a congruent number. %H A003273 G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. %H A003273 G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy] %H A003273 Allan J. MacLeod, The congruent number descent of Komotu, Watanabe and Wada, arXiv:2005.02615 [math.NT], 2020. %H A003273 MathDL, Five Mathematicians Capture Record Number of Congruent Numbers %H A003273 Fidel Ronquillo Nemenzo, All congruent numbers less than 40000, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31. %H A003273 Karl Rubin, Right triangles and elliptic curves %H A003273 W. A. Stein, Introduction to the Congruent Number Problem %H A003273 W. A. Stein, The Congruent Number Problem %H A003273 Ye Tian, Congruent Numbers and Heegner Points, arXiv:1210.8231 [math.NT], 2012. %H A003273 J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334. %H A003273 D. J. Wright, The Congruent Number Problem %e A003273 24 is congruent because 24 is the area of the right triangle with sides 6,8,10. %e A003273 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 %t A003273 (* 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 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. A006991, A072068, A072069, A072070, A072071, A073120, A165564, A182429, A256418, A259680-A259687. %K A003273 nonn,nice %O A003273 1,1 %A A003273 _N. J. A. Sloane_ %E A003273 Guy gives a table up to 1000. %E A003273 Edited by _T. D. Noe_, Jun 14 2002 %E A003273 Comments revised by _Max Alekseyev_, Nov 15 2008 %E A003273 Comment corrected by _Jonathan Sondow_, Oct 10 2013 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE