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).
From Jose Aranda, Jul 04 2024: (Start)
The C++ program I have written calculates a(8) = 31925924 in 75 minutes. The time grows almost exponentially.
Looking at the 8 known terms I think the above conjecture should perhaps refer to A274264 rather than to the present sequence.
From the link "A trillion triangles": "The calculation found of these most mysterious congruent numbers up to a trillion = 3148379694."
That number corresponds to a(10) = 108744287 + A274264(10).
With A274264(10) = 3039635407. Now
3/Pi^2 = 0.303963550927013314...
A274264(08) = 0030396356.
A274264(10) = 003039635407.
A274264(18) = 00303963550927001730.
The sequence A274264 tends to this limit. This sequence may not. (End)
LINKS
Jose Aranda, C++ program
Estelle Basor and Bill Hart, A trillion triangles, American Institute of Mathematics,
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
KEYWORD
nonn,hard,more
AUTHOR
Frank M Jackson, Jun 20 2016
EXTENSIONS
a(7) corrected by Frank M Jackson, Jul 25 2016
a(8) from Jose Aranda, Jul 04 2024
STATUS
approved