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 A274403 Number of primitive (squarefree) congruent numbers (A006991) <= 10^n. 0
 3, 36, 361, 3503, 34065, 332712, 3252966 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: the limit of a(n)/10^n tends to 3/Pi^2 (A104141). This is based on the assumption, conditional on the Birch Swinnerton-Dyer conjecture, that all squarefree integers congruent to {5, 6, 7} mod 8 (A273929) are a subset of primitive congruent numbers (A006991) and have a natural density of 3/Pi^2. However, squarefree integers congruent to {1, 2, 3} mod 8 are conjecturally sparsely congruent numbers with a natural density of 0. It has been proved without the BSD conjecture that the natural density of congruent numbers is at least 55.9% the natural density of squarefree numbers congruent to {5, 6, 7} mod 8 (see A. Smith link). The Mathematica program below is a slow implementation of the Tunnell criteria for determining congruent numbers. It will give counts for up to 10^5 in realistic time. Counts for 10^6 and 10^7 have been derived from tables generated by Giovanni Resta (see link). LINKS Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, (2008). Giovanni Resta, Table of primitive congruent numbers {1, 2, 3} mod 8 Alexander Smith, The congruent numbers have positive natural density, arXiv:1603.08479 [math.NT], 2016. Wikipedia, Congruent number Shou-Wu Zhang, The Congruent Numbers and Heegner Points, Asian Pacific Mathematics Newsletter, Vol 3(2) (2013). MATHEMATICA 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 + 2 y^2 + 8 z^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+4y^2+32z^2==n/2, {x, y, z}, Integers], ok=True]; ok)]; Table[Length@Select[Range[10^n], CongruentQ], {n, 1, 5}] CROSSREFS Cf. A006991, A062695, A071172, A104141, A273929, A274043, A274264. Sequence in context: A092648 A026121 A055303 * A068177 A249894 A099670 Adjacent sequences:  A274400 A274401 A274402 * A274404 A274405 A274406 KEYWORD nonn,more AUTHOR Frank M Jackson, Jun 20 2016 EXTENSIONS a(7) corrected by Frank M Jackson, Jul 25 2016 STATUS approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)