Search: id:A006991
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%I A006991 M3748
%S A006991 5,6,7,13,14,15,21,22,23,29,30,31,34,37,38,39,41,46,47,53,55,61,62,
%T A006991 65,69,70,71,77,78,79,85,86,87,93,94,95,101,102,103,109,110,111,118,
%U A006991 119,127,133,134,137,138,141,142,143,145,149,151,154,157,158,159
%N A006991 Primitive congruent numbers.
%C A006991 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 (noe(AT)sspectra.com), Jun 13 2002
%D A006991 R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 
               (1974), 303-305 and 30 (1976), 198.
%D A006991 R. K. Guy, Unsolved Problems in Number Theory, D27.
%D A006991 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006991 J. B. Tunnell, A classical Diophantine problem and modular forms of weight 
               3/2, Invent. Math., 72 (1983), 323-334.
%H A006991 T. D. Noe, <a href="b006991.txt">Primitive congruent numbers up to 10000; 
               table of n, a(n) for n = 1..3503</a>
%H A006991 American Institute of Mathematics, <a href="http://www.aimath.org/news/
               congruentnumbers/">A trillion triangles</a>
%H A006991 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 A006991 Clay Mathematics Institute, <a href="http://www.claymath.org/prizeproblems/
               birchsd.htm">The Birch and Swinnerton-Dyer Conjecture</a>
%H A006991 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 A006991 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               math10/matb2000.htm">361 Congruent Numbers g: 1<=g<=999</a>
%H A006991 Karl Rubin, <a href="http://math.Stanford.EDU/~rubin/lectures/sumo/">
               Elliptic curves and right triangles</a>
%e A006991 6 is congruent because 6 is the area of the right triangle with sides 
               3,4,5. It is primitive because it is square-free.
%t A006991 The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer 
               conjecture and uses functions from A072068.
%t A006991 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
%Y A006991 Cf. A003273, A072068, A072069, A072070, A072071.
%Y A006991 Sequence in context: A106745 A165776 A003273 this_sequence A047574 A067531 
               A031029
%Y A006991 Adjacent sequences: A006988 A006989 A006990 this_sequence A006992 A006993 
               A006994
%K A006991 nonn
%O A006991 1,1
%A A006991 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A006991 More terms from T. D. Noe, Feb 26, 2003

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