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A006991 Primitive congruent numbers.
(Formerly M3748)
9
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 Swinnerton-Dyer 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), 303-305 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 Swinnerton-Dyer Conjecture

Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem

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), 323-334.

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 Swinnerton-Dyer 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

CROSSREFS

Cf. A003273, A072068, A072069, A072070, A072071.

Sequence in context: A106745 A165776 A003273 * A047574 A067531 A031029

Adjacent sequences:  A006988 A006989 A006990 * A006992 A006993 A006994

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified October 21 14:25 EDT 2014. Contains 248377 sequences.