1,1

Positive integers n such that x^2 + n*y^2 = z^2 and x^2 - n*y^2 = t^2 have simultaneous integer solutions. In other words, n is the difference of an arithmetic progression of three rational squares: (t/y)^2, (x/y)^2, (z/y)^2. Values of n corresponding to y=1 (i.e., an arithmetic progression of three integer squares) form A057102.

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.

Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 23 2009]

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.

H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 454. [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 23 2009]

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

American Institute of Mathematics, A trillion triangles

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

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

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

Bill Hart, A Trillion Triangles [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Oct 01 2009]

Author?, Five Mathematicians Capture Record Number of Congruent Numbers [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Oct 01 2009]

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. A057102, A006991, A072068, A072069, A072070, A072071.

Sequence in context: A011761 A106745 A165776 this_sequence A006991 A047574 A067531

Adjacent sequences: A003270 A003271 A003272 this_sequence A003274 A003275 A003276

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Guy gives a table up to 1000.

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

Comments revised by Max Alekseyev (maxale(AT)gmail.com), Nov 15 2008