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A006991
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Primitive congruent numbers.
(Formerly M3748)
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8
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5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 34, 37, 38, 39, 41, 46, 47, 53, 55, 61, 62, 65, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95, 101, 102, 103, 109, 110, 111, 118, 119, 127, 133, 134, 137, 138, 141, 142, 143, 145, 149, 151, 154, 157, 158, 159
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OFFSET
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1,1
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COMMENTS
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Assuming the Birch and Swinnerton-Dyer conjecture, determining whether a 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. The Mathematica program for this sequence uses variables defined in A072068, A072069, A072070, A072071. - T. D. Noe, Jun 13 2002
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REFERENCES
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R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.
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).
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
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LINKS
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T. D. Noe, Primitive congruent numbers up to 10000; table of n, a(n) for n = 1..3503
American Institute of Mathematics, A trillion triangles
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Hisanori Mishima, 361 Congruent Numbers g: 1<=g<=999
Karl Rubin, Elliptic curves and right triangles
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EXAMPLE
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6 is congruent because 6 is the area of the right triangle with sides 3,4,5. It is primitive because it is squarefree.
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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 functions from A072068.
For[lst={}; n=1, n<=maxN, n++, If[SquareFreeQ[n], If[(EvenQ[n]&&soln3[[n/2]]==2soln4[[n/2]])|| (OddQ[n]&&soln1[[(n+1)/2]]==2soln2[[(n+1)/2]]), AppendTo[lst, n]]]]; lst
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CROSSREFS
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Cf. A003273, A072068, A072069, A072070, A072071.
Sequence in context: A106745 A165776 A003273 * A047574 A067531 A031029
Adjacent sequences: A006988 A006989 A006990 * A006992 A006993 A006994
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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EXTENSIONS
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More terms from T. D. Noe, Feb 26, 2003
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STATUS
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approved
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