cp's OEIS Frontend

A006991 Primitive congruent numbers.

%I A006991 M3748 #108 Aug 06 2024 13:03:56
%S A006991 5,6,7,13,14,15,21,22,23,29,30,31,34,37,38,39,41,46,47,53,55,61,62,65,
%T A006991 69,70,71,77,78,79,85,86,87,93,94,95,101,102,103,109,110,111,118,119,
%U A006991 127,133,134,137,138,141,142,143,145,149,151,154,157,158,159
%N A006991 Primitive congruent numbers.
%C A006991 Squarefree terms of A003273.
%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_, Jun 13 2002
%D A006991 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 155.
%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).
%H A006991 T. D. Noe, <a href="/A006991/b006991.txt">Primitive congruent numbers up to 10000; table of n, a(n) for n = 1..3503</a>
%H A006991 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 A006991 American Institute of Mathematics, <a href="http://www.aimath.org/news/congruentnumbers/">A trillion triangles</a>
%H A006991 Jose Aranda, <a href="/A006991/a006991.cpp.txt">C++ program</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 Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/articles/congruentnumber.pdf">The Congruent Number Problem</a>, The Harvard College Mathematics Review, 2008.
%H A006991 Department of Pure Maths., Univ. Sheffield, <a href="https://web.archive.org/web/20040206183520/http://www.shef.ac.uk/~puremath/theorems/congruent.html">Pythagorean triples and the congruent number problem</a>
%H A006991 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 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 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/congruent_number/">Congruent numbers</a> Primitive congruent numbers up to 10^7.
%H A006991 Karl Rubin, <a href="http://math.Stanford.EDU/~rubin/lectures/sumo/">Elliptic curves and right triangles</a>
%H A006991 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 A006991 Wikipedia, <a href="https://en.wikipedia.org/wiki/Congruent_number">Congruent number</a>
%H A006991 R. G. Wilson v, <a href="/A006991/a006991.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>
%e A006991 6 is congruent because 6 is the area of the right triangle with sides 3,4,5. It is a primitive congruent number because it is squarefree.
%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
%t A006991 (* The following self-contained Mathematica code also assumes the truth of the Birch and Swinnerton-Dyer conjecture. *)
%t A006991 CongruentQ[n_] := Module[{x, y, z, ok=False}, (Which[! SquareFreeQ[n], Null[], MemberQ[{5, 6, 7}, Mod[n, 8]], ok = True, OddQ@n&&Length@Solve[x^2+2y^2+8z^2==n, {x, y, z}, Integers]==2Length@Solve[x^2+2y^2+32z^2==n, {x, y, z}, Integers], ok=True, EvenQ@n&&Length@Solve[x^2+4y^2+8z^2==n/2, {x, y, z}, Integers]==2Length@ Solve[x^2 + 4 y^2 + 32 z^2 == n/2, {x, y, z}, Integers], ok=True]; ok)]; Select[Range[200], CongruentQ] (* _Frank M Jackson_, Jun 06 2016 *)
%Y A006991 Cf. A003273, A072068, A072069, A072070, A072071.
%K A006991 nonn
%O A006991 1,1
%A A006991 _N. J. A. Sloane_, _Robert G. Wilson v_
%E A006991 More terms from _T. D. Noe_, Feb 26 2003