1,1

n such that x^2 + n*y^2 = z^2 and x^2 - n*y^2 = t^2 have simultaneous integer solutions.

Tunnell shows that if a number is square-free and congruent, then the ratio of the number of solutions of a pair of equations is 2. If the Birch and Swinnerton-Dyer conjecture is assumed, then determining whether a square-free 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. If a number n is congruent, there are an infinite number of right triangles having rational sides and area n. All congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a square number A000290.

The Mathematica program for this sequence uses the list of primitive congruent numbers produced by the Mathematica program in A006991.

R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 43-45.

R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.

R. Cuculiere, "Mille ans de chasse aux nombres congruents", in Pour la Science (French edition of 'Scientific American'), No. 7, 1987, pp. 14-18.

L. E. Dickson, History of the Theory of Numbers, Vol. 2, pp. 459-472, AMS Chelsea Pub. Providence RI 1999.

R. K. Guy, Unsolved Problems in Number Theory, D27.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.

J. B. Tunnell, A classical diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

T. D. Noe, Congruent numbers up to 10000; table of n, a(n) for n = 1..5742

E. Brown, Three Fermat Trails to Elliptic Curves, 5. Congruent Numbers and Elliptic Curves (pp 8-11/17)

Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture

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

E. V. Eikenberg, The Congruent Number Problem

W. F. Hammond, A Reading of Karl Rubin's SumO Slides on Rational Right Triangles and Elliptic Curves

Karl Rubin, Elliptic curves and right triangles

W. A. Stein, Introduction to the Congruent Number Problem

W. A. Stein, The Congruent Number Problem

D. J. Wright, The Congruent Number Problem

24 is congruent because 24 is the area of the right triangle with sides 6,8,10.

The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer conjecture:

For[cLst={}; i=1, i<=Length[lst], i++, n=lst[[i]]; j=1; While[n j^2<=maxN, cLst=Union[cLst, {n j^2}]; j++ ]]; cLst

Cf. A006991, A072068, A072069, A072070, A072071.

Sequence in context: A047320 A011761 A106745 this_sequence A006991 A047574 A067531

Adjacent sequences: A003270 A003271 A003272 this_sequence A003274 A003275 A003276

nonn,nice

njas

Guy gives a table up to 1000.

Edited by T. D. Noe (noe(AT)sspectra.com), Jun 14 2002