

A006991


Primitive congruent numbers.
(Formerly M3748)


21



5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 34, 37, 38, 39, 41, 46, 47, 53, 55, 61, 62, 65, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95, 101, 102, 103, 109, 110, 111, 118, 119, 127, 133, 134, 137, 138, 141, 142, 143, 145, 149, 151, 154, 157, 158, 159
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Assuming the Birch and SwinnertonDyer 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


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, D27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Primitive congruent numbers up to 10000; table of n, a(n) for n = 1..3503
R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303305 and 30 (1976), 198.
American Institute of Mathematics, A trillion triangles
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009
Clay Mathematics Institute, The Birch and SwinnertonDyer Conjecture
Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
A. Dujella, A. S.Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8
Hisanori Mishima, 361 Congruent Numbers g: 1<=g<=999
Giovanni Resta, Congruent numbers Primitive congruent numbers up to 10^7
Karl Rubin, Elliptic curves and right triangles
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323334.


EXAMPLE

6 is congruent because 6 is the area of the right triangle with sides 3,4,5. It is primitive because it is squarefree.


MATHEMATICA

The following Mathematica code assumes the truth of the Birch and SwinnertonDyer conjecture and uses functions from A072068.
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
The following selfcontained Mathematica code also assumes the truth of the Birch and SwinnertonDyer conjecture.
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 *)


CROSSREFS

Cf. A003273, A072068, A072069, A072070, A072071.
Sequence in context: A106745 A165776 A003273 * A047574 A273929 A067531
Adjacent sequences: A006988 A006989 A006990 * A006992 A006993 A006994


KEYWORD

nonn,changed


AUTHOR

N. J. A. Sloane, Robert G. Wilson v


EXTENSIONS

More terms from T. D. Noe, Feb 26 2003


STATUS

approved



