A006991 -id:A006991 - cp's OEIS frontend

A274403 Number of primitive (squarefree) congruent numbers (A006991) <= 10^n.

3, 36, 361, 3503, 34065, 332712, 3252966, 31925924

Offset: 1

Frank M Jackson, Jun 20 2016

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)

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).

Cf. A006991, A062695, A071172, A104141, A273929, A274043, A274264.

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}]

a(7) corrected by Frank M Jackson, Jul 25 2016

a(8) from Jose Aranda, Jul 04 2024

A242061 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1, x+y odd and for the least y, {x, y} are integers such that x*y(y^2-x^2)/A006991(n) is a perfect square.

Original entry on oeis.org

14, 2, 129, 52686, 29, 7, 9, 1274, 296125969, 12012350, 5, 1279281, 44, 302583265614, 780914, 90, 316, 2605, 106023820090609, 1754402265205275806, 7794, 72957466300254, 768323201, 40, 18505, 23, 6478321, 3966329, 326, 14280500082452241

Offset: 1

Frank M Jackson, Aug 13 2014

nonn

The triangle array A226314(n)/A054531(n) that enumerates all positive rationals x/y can be generalized to enumerate all ordered pairs {x, y} where x and y are natural numbers. For example, A243808 uses a subset of this triangular array to enumerate all primitive Pythagorean triples (PPT).

A006991(n) is the sequence of primitive congruent numbers and by definition there exists a PPT whose area is equal to k^2*A006991(n) for some integer k. a(n) is an enumeration of these PPT's and is a measure of the number of Pythagorean triangles that have to be searched to find a PPT with the least hypotenuse that has an area equal to k^2*A006991(n). If {x, y} are the generators of a PPT (a, b, c) where a = y^2-x^2, b = 2x*y, c=y^2+x^2 then its area = x*y(y^2-x^2). The Mathematica program limits searches to the first 12.5 million generated PPT's. All other results have been obtained from tables cataloged by Hisanori Mishima (see Links).

			.  j       {A226314(n),A054531(n)}, 1<=i<=j<=12 and n=i+j(j-1)/2
.  --   --------------------------------------------------------
.   1:  1,1
.   2:  1,2 2,1
.   3:  1,3 2,3 3,1
.   4:  1,4 3,2 3,4 4,1
.   5:  1,5 2,5 3,5 4,5 5,1
.   6:  1,6 4,3 5,2 5,3 5,6 6,1
.   7:  1,7 2,7 3,7 4,7 5,7 6,7 7,1
.   8:  1,8 5,4 3,8 7,2 5,8 7,4 7,8 8,1
.   9:  1,9 2,9 7,3 4,9 5,9 8,3 7,9 8,9 9,1
.  10:  1,10 6,5 3,10 7,5 9,2 8,5 7,10 9,5 9,10 10,1
.  11:  1,11 2,11 3,11 4,11 5,11 6,11 7,11 8,11 9,11 10,11 11,1
.  12:  1,12 7,6 9,4 10,3 5,12 11,2 7,12 11,3 11,4 11,6 11,12 12,1 .
a(13)=44 and A006991(13)=34 so 34 is the 13th congruent number. a(13) gives the 44th term of the triangular array at index (8, 9). This gives (x, y) as (8, 9), it generates the PPT (17, 144, 145) and has an area 6^2*34 = 1224.

Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
Hisanori Mishima, 361 Congruent Numbers g: 1<=g<=999, Mathematician's Secret Room, Chapter 10 : Congruent Numbers (D27 Congruent numbers).
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A006991, A054531, A226314.

lst1={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}; getpos[n0_] := (lst=0; Do[If[IntegerQ@Sqrt[m*n(m-n)(m+n)/n0]&&OddQ[m+n] && GCD[m, n]==1, (lst=m(m-1)/2+n; Break[])], {m, 2, 5000}, {n, 1, m-1}]; lst); SetAttributes[getpos, Listable]; getpos[lst1]

A003273 Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.

Original entry on oeis.org

5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126

Offset: 1

N. J. A. Sloane

Positive integers k such that x^2 + k*y^2 = z^2 and x^2 - k*y^2 = t^2 have simultaneous integer solutions. In other words, k is the difference of an arithmetic progression of three rational squares: (t/y)^2, (x/y)^2, (z/y)^2. Values of k corresponding to y=1 (i.e., an arithmetic progression of three integer squares) form A256418.
Tunnell shows that if a number is squarefree 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 squarefree number k is congruent requires counting the solutions to a pair of equations. For odd k, see A072068 and A072069; for even k see A072070 and A072071.
If a number k is congruent, there are an infinite number of right triangles having rational sides and area k. All congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a positive square number A000290.
Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. - Steven Finch, Apr 23 2009

			24 is congruent because 24 is the area of the right triangle with sides 6,8,10.
5 is congruent because 5 is the area of the right triangle with sides 3/2, 20/3, 41/6 (although not of any right triangle with integer sides -- see A073120). - _Jonathan Sondow_, Oct 04 2013

Alter, Ronald; Curtz, Thaddeus B.; Kubota, K. K. Remarks and results on congruent numbers. Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1972), pp. 27-35. Florida Atlantic Univ., Boca Raton, Fla., 1972. MR0349554 (50 #2047)
H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 454. [From Steven Finch, Apr 23 2009]
R. Cuculière, "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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Congruent numbers up to 10000; table of n, a(n) for n = 1..5742
R. Alter, Letter to N. J. A. Sloane, Sep 1975
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.
A. Alvarado and E. H. Goins, Arithmetic progressions on conic sections, arXiv:1210.6612 [math.NT], 2012. [From _Jonathan Sondow_, Oct 25 2012]
Estelle Basor and Bill Hart, A trillion triangles, American Institute of Mathematics,
E. Brown, Three Fermat Trails to Elliptic Curves, 5. Congruent Numbers and Elliptic Curves (pp 8-11/17)
Graeme Brown, The Congruent Number Problem, 2014.
Jasbir S. Chahal, Some remarks on rational right triangles, Expos. Math. (2024).
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009.
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Raiza Corpuz, Constructing congruent number elliptic curves using 2-descent, arXiv:2006.08113 [math.NT], 2020.
R. Cuculière, Mille ans de chasse aux nombres congruents, Séminaire de Philosophie et Mathématiques, 2, 1988, p. 1-17.
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem [Cached copy]
A. Dujella, A. S.Janfeda, and S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8.
E. V. Eikenberg, The Congruent Number Problem
David Goldberg, Triangle Sides for Congruent Numbers less than 10000, arXiv:2106.07373 [math.NT], 2021.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent number elliptic curves with rank at least two, arXiv:1809.02037 [math.NT], 2018.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent Number Elliptic Curves Related to Integral Solutions of m^2 = n^2 + nl + n^2, J. Int. Seq., Vol. 22 (2019), Article 19.3.1.
Alvaro Lozano-Robledo, My #MegaFavNumber: 224,403,517,704,336,969,924,557,513,090,674,863,160,948,472,041, video (2020) [discusses congruent numbers and a(157)]
W. F. Hammond, A Reading of Karl Rubin's SUMO Slides on Rational Right Triangles and Elliptic Curves
Bill Hart, A Trillion Triangles, American Institute of Mathematics.
T. Komatsu, Congruent numbers and continued fractions, Fib. Quart., 50 (2012), 222-226. - From _N. J. A. Sloane_, Mar 04 2013
S. Komoto, T. Watanabe and H. Wada, 42553 is a congruent number.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]
Allan J. MacLeod, The congruent number descent of Komotu, Watanabe and Wada, arXiv:2005.02615 [math.NT], 2020.
MathDL, Five Mathematicians Capture Record Number of Congruent Numbers
Fidel Ronquillo Nemenzo, All congruent numbers less than 40000, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31.
Karl Rubin, Right triangles and elliptic curves
W. A. Stein, Introduction to the Congruent Number Problem
W. A. Stein, The Congruent Number Problem
Ye Tian, Congruent Numbers and Heegner Points, arXiv:1210.8231 [math.NT], 2012.
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
D. J. Wright, The Congruent Number Problem

Cf. A006991, A072068, A072069, A072070, A072071, A073120, A165564, A182429, A256418, A259680-A259687.

(* The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer conjecture and uses the list of primitive congruent numbers produced by the Mathematica code in A006991: *)
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

Guy gives a table up to 1000.
Edited by T. D. Noe, Jun 14 2002
Comments revised by Max Alekseyev, Nov 15 2008
Comment corrected by Jonathan Sondow, Oct 10 2013

A024406 Ordered areas of primitive Pythagorean triangles.

Original entry on oeis.org

6, 30, 60, 84, 180, 210, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 7980, 8970, 8976, 9690

Offset: 1

David W. Wilson

This sequence also gives Fibonacci's congruous numbers (or congrua) divided by 4 with multiplicities, not regarding leg exchange in the underlying primitive Pythagorean triangle. See A258150 and the example. - Wolfdieter Lang, Jun 14 2015

The squarefree part of an entry which is not squarefree is a primitive congruent number from A006991 belonging to a Pythagorean triangle with rational (not all integer) side lengths (and its companion obtained by exchanging the legs). See the W. Lang link. - Wolfdieter Lang, Oct 25 2016

			a(6) = a(7) = 210 corresponds to the area (in some squared length unit) of the primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). Fibonacci's congruum C = 840 = 210*4 belongs to the two triples [x, y, z] = [29, 41, 1] and [37, 47, 23], solving x^2 + C = y^2 and x^2 - C = z^2. - _Wolfdieter Lang_, Jun 14 2015
a(5) = 180 = 6^2*5 lead to the primitive congruent number A006991(1) = 5 from the primitive Pythagorean triangle [9, 40, 41] after division by 6: [3/2, 20/3, 41/6]. See the link for the other nonsquarefree a(n) numbers. - _Wolfdieter Lang_, Oct 25 2016

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators
Wolfdieter Lang, Non-squarefree entries, their congruent numbers and rational Pythagorean triangles
Eric Weisstein's World of Mathematics, Congruum Problem

Cf. A009112, A024365, A094182, A094183, A256418 (congrua), A258150.

a(n) = 6*A020885(n). - Lekraj Beedassy, Apr 30 2004

a(n) = A121728(n)*A121729(n)/2. - M. F. Hasler, Apr 16 2020

A062695 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 2.

Original entry on oeis.org

34, 41, 65, 137, 138, 145, 154, 161, 194, 210, 219, 226, 257, 265, 291, 299, 313, 323, 330, 353, 371, 386, 395, 410, 426, 434, 442, 457, 465, 505, 514, 546, 561, 602, 609, 651, 658, 674, 689, 721, 723, 731, 761, 777, 793, 866, 889, 890, 905, 915, 985, 987, 995

Offset: 1

Noam D. Elkies, Jul 04 2001

nonn

These n are precisely the primitive congruent numbers (A006991) with n==1, n==2, or n==3 (mod 8). - T. D. Noe, Aug 02 2006

Jinyuan Wang, Table of n, a(n) for n = 1..453
A. Dujella, A. S. Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8
N. D. Elkies, Algorithmic (a.k.a. Computational) Number Theory: Tables, Links, etc.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]
Kazunari Noda and Hideo Wada, All congruent numbers less than 10000, Proc. Japan Acad. Ser. A Math. Sci., Volume 69, Number 6 (1993), 175-178.

Cf. A003273, A006991, A062693, A062694, A259680-A259687.

r(n)=ellanalyticrank(ellinit([0,0,0,-n^2,0]))[1]
for(n=1,1e3,if(issquarefree(n)&&r(n)==2,print1(n", "))) \\ Charles R Greathouse IV, Sep 01 2011; corrected by Frank M Jackson, Aug 04 2016

More terms from Jinyuan Wang, Dec 12 2020

cp's OEIS Frontend

A364108 a(n) is the larger coefficient of the pair (x, y) such that (x^2-y^2)/r, 2*x*y/r, (x^2+y^2)/r are the 2 legs and hypotenuse of the least Pythagorean triple having area A006991(n).

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A364109 a(n) is the lesser coefficient of the pair (x, y) such that (x^2-y^2)/r, 2*x*y/r, (x^2+y^2)/r are the 2 legs and hypotenuse of the least Pythagorean triple having area A006991(n).

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A364110 a(n) = sqrt((x^2 - y^2)*x*y/c) where x is A364108(n), y is A364109(n) and c is A006991(n).

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A274403 Number of primitive (squarefree) congruent numbers (A006991) <= 10^n.

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Mathematica

Extensions

A242061 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1, x+y odd and for the least y, {x, y} are integers such that x*y(y^2-x^2)/A006991(n) is a perfect square.

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A003273 Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.

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Mathematica

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A024406 Ordered areas of primitive Pythagorean triangles.

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A062695 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 2.

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A259680 Let m = A062695(n); a(n) is value of s in decomposition of m defined in Comments.

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A259687 Let m = A062695(n); a(n) is value of d in decomposition of m defined in Comments.

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A364108 a(n) is the larger coefficient of the pair (x, y) such that (x^2-y^2)/r, 2xy/r, (x^2+y^2)/r are the 2 legs and hypotenuse of the least Pythagorean triple having area A006991(n).

A364109 a(n) is the lesser coefficient of the pair (x, y) such that (x^2-y^2)/r, 2xy/r, (x^2+y^2)/r are the 2 legs and hypotenuse of the least Pythagorean triple having area A006991(n).

A364110 a(n) = sqrt((x^2 - y^2)xy/c) where x is A364108(n), y is A364109(n) and c is A006991(n).