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A274403
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Number of primitive (squarefree) congruent numbers (A006991) <= 10^n.
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0
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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Table of n, a(n) for n=1..7.
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).
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MATHEMATICA
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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}]
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CROSSREFS
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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
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KEYWORD
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nonn,more
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AUTHOR
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Frank M Jackson, Jun 20 2016
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EXTENSIONS
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a(7) corrected by Frank M Jackson, Jul 25 2016
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STATUS
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approved
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