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A274403 Number of primitive (squarefree) congruent numbers (A006991) <= 10^n.

%I A274403 #71 Jul 19 2024 22:23:25
%S A274403 3,36,361,3503,34065,332712,3252966,31925924
%N A274403 Number of primitive (squarefree) congruent numbers (A006991) <= 10^n.
%C A274403 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).
%C A274403 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).
%C A274403 From _Jose Aranda_, Jul 04 2024: (Start)
%C A274403 The C++ program I have written calculates a(8) = 31925924 in 75 minutes. The time grows almost exponentially.
%C A274403 Looking at the 8 known terms I think the above conjecture should perhaps refer to A274264 rather than to the present sequence.
%C A274403 From the link "A trillion triangles": "The calculation found of these most mysterious congruent numbers up to a trillion = 3148379694."
%C A274403 That number corresponds to a(10) = 108744287 + A274264(10).
%C A274403 With A274264(10) = 3039635407. Now
%C A274403 3/Pi^2      = 0.303963550927013314...
%C A274403 A274264(08) = 0030396356.
%C A274403 A274264(10) = 003039635407.
%C A274403 A274264(18) = 00303963550927001730.
%C A274403 The sequence A274264 tends to this limit. This sequence may not. (End)
%H A274403 Jose Aranda, <a href="/A274403/a274403.cpp.txt">C++ program</a>
%H A274403 Estelle Basor and Bill Hart, <a href="https://www.aimath.org/news/congruentnumbers">A trillion triangles</a>, American Institute of Mathematics,
%H A274403 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 A274403 Giovanni Resta, <a href="http://www.numbersaplenty.com/data/primitive-congruent-mod123.txt">Table of primitive congruent numbers {1, 2, 3} mod 8</a>
%H A274403 Alexander Smith, <a href="http://arxiv.org/abs/1603.08479">The congruent numbers have positive natural density</a>, arXiv:1603.08479 [math.NT], 2016.
%H A274403 Wikipedia, <a href="http://en.wikipedia.org/wiki/Congruent_number">Congruent number</a>
%H A274403 Shou-Wu Zhang, <a href="http://www.asiapacific-mathnews.com/03/0302/0012_0015.pdf">The Congruent Numbers and Heegner Points</a>, Asian Pacific Mathematics Newsletter, Vol 3(2) (2013).
%t A274403 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}]
%Y A274403 Cf. A006991, A062695, A071172, A104141, A273929, A274043, A274264.
%K A274403 nonn,hard,more
%O A274403 1,1
%A A274403 _Frank M Jackson_, Jun 20 2016
%E A274403 a(7) corrected by _Frank M Jackson_, Jul 25 2016
%E A274403 a(8) from _Jose Aranda_, Jul 04 2024