%I M3747 #142 Oct 28 2024 17:09:10

%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 k for which there exists a right triangle having area k and rational sides.

%C 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 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 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 Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. - _Steven Finch_, Apr 23 2009

%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. Cuculière, "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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%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 R. Alter, <a href="/A003273/a003273_1.pdf">Letter to N. J. A. Sloane, Sep 1975</a>

%H R. Alter, <a href="http://www.jstor.org/stable/2320381">The congruent number problem</a>, Amer. Math. Monthly, 87 (1980), 43-45.

%H R. Alter and T. B. Curtz, <a href="http://www.jstor.org/stable/2005838">A note on congruent numbers</a>, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.

%H A. Alvarado and E. H. Goins, <a href="http://arxiv.org/abs/1210.6612">Arithmetic progressions on conic sections</a>, arXiv:1210.6612 [math.NT], 2012. [From _Jonathan Sondow_, Oct 25 2012]

%H Estelle Basor and Bill Hart, <a href="https://www.aimath.org/news/congruentnumbers">A trillion triangles</a>, American Institute of Mathematics,

%H E. Brown, Three Fermat Trails to Elliptic Curves, <a href="https://intranet.math.vt.edu/people/brown/doc/ellip.pdf">5. Congruent Numbers and Elliptic Curves (pp 8-11/17)</a>

%H Graeme Brown, <a href="http://www.graemebrownart.com/congruent.pdf">The Congruent Number Problem</a>, 2014.

%H Jasbir S. Chahal, <a href="https://doi.org/10.1016/j.exmath.2024.125623">Some remarks on rational right triangles</a>, Expos. Math. (2024).

%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/millenium-problems/birch-and-swinnerton-dyer-conjecture">The Birch and Swinnerton-Dyer Conjecture</a>

%H Raiza Corpuz, <a href="https://arxiv.org/abs/2006.08113">Constructing congruent number elliptic curves using 2-descent</a>, arXiv:2006.08113 [math.NT], 2020.

%H R. Cuculière, <a href="http://www.numdam.org/item/SPHM_1988___2_A1_0">Mille ans de chasse aux nombres congruents</a>, Séminaire de Philosophie et Mathématiques, 2, 1988, p. 1-17.

%H Department of Pure Maths., Univ. Sheffield, <a href="https://web.archive.org/web/20040206183520/http://www.shef.ac.uk:80/~puremath/theorems/congruent.html">Pythagorean triples and the congruent number problem</a> [Cached copy]

%H A. Dujella, A. S.Janfeda, and S. Salami, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janfada/janfada3.html">A Search for High Rank Congruent Number Elliptic Curves</a>, JIS 12 (2009) 09.5.8.

%H E. V. Eikenberg, <a href="http://www.uwyo.edu/bshader/mathtlcalgebra/congruentnumbers.pdf">The Congruent Number Problem</a>

%H David Goldberg, <a href="https://arxiv.org/abs/2106.07373">Triangle Sides for Congruent Numbers less than 10000</a>, arXiv:2106.07373 [math.NT], 2021.

%H Lorenz Halbeisen and Norbert Hungerbühler, <a href="https://arxiv.org/abs/1809.02037">Congruent number elliptic curves with rank at least two</a>, arXiv:1809.02037 [math.NT], 2018.

%H Lorenz Halbeisen and Norbert Hungerbühler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Halbeisen/halb4.html"> Congruent Number Elliptic Curves Related to Integral Solutions of m^2 = n^2 + nl + n^2</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.1.

%H Alvaro Lozano-Robledo, <a href="https://www.youtube.com/watch?v=Q-tNzDFnzKI">My #MegaFavNumber: 224,403,517,704,336,969,924,557,513,090,674,863,160,948,472,041</a>, video (2020) [discusses congruent numbers and a(157)]

%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>, American Institute of Mathematics.

%H T. Komatsu, <a href="https://www.fq.math.ca/Papers1/50-3/KomatsuCongruentNmbrs.pdf">Congruent numbers and continued fractions</a>, Fib. Quart., 50 (2012), 222-226. - From _N. J. A. Sloane_, Mar 04 2013

%H S. Komoto, T. Watanabe and H. Wada, <a href="http://jant.jsiam.org/pub/ac05/Wada/wada.pdf">42553 is a congruent number</a>.

%H G. Kramarz, <a href="http://dx.doi.org/10.1007/BF01451411">All congruent numbers less than 2000</a>, Math. Annalen, 273 (1986), 337-340.

%H G. Kramarz, <a href="/A003273/a003273.pdf">All congruent numbers less than 2000</a>, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

%H Allan J. MacLeod, <a href="https://arxiv.org/abs/2005.02615">The congruent number descent of Komotu, Watanabe and Wada</a>, arXiv:2005.02615 [math.NT], 2020.

%H MathDL, <a href="http://mathdl.maa.org/mathDL/?pa=mathNews&amp;sa=view&amp;newsId=680">Five Mathematicians Capture Record Number of Congruent Numbers</a>

%H Fidel Ronquillo Nemenzo, <a href="https://doi.org/10.3792/pjaa.74.29">All congruent numbers less than 40000</a>, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31.

%H Karl Rubin, <a href="http://www.math.uci.edu/~krubin/lectures/rossweb.pdf">Right triangles and elliptic curves</a>

%H W. A. Stein, <a href="http://wstein.org/edu/Fall2001/124/lectures/lecture33/html">Introduction to the Congruent Number Problem</a>

%H W. A. Stein, <a href="http://wstein.org/simuw06">The Congruent Number Problem</a>

%H Ye Tian, <a href="http://arxiv.org/abs/1210.8231">Congruent Numbers and Heegner Points</a>, arXiv:1210.8231 [math.NT], 2012.

%H J. B. Tunnell, <a href="http://dx.doi.org/10.1007/BF01389327">A classical Diophantine problem and modular forms of weight 3/2</a>, Invent. Math., 72 (1983), 323-334.

%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.

%e 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 (* 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. A006991, A072068, A072069, A072070, A072071, A073120, A165564, A182429, A256418, A259680-A259687.

%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

%E Comment corrected by _Jonathan Sondow_, Oct 10 2013