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A003273
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Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.
(Formerly M3747)
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43
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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
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OFFSET
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1,1
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COMMENTS
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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.
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.
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.
Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. - Steven Finch, Apr 23 2009
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REFERENCES
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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).
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LINKS
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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)]
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EXAMPLE
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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
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MATHEMATICA
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(* 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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Guy gives a table up to 1000.
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STATUS
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approved
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