A003273 -id:A003273 - cp's OEIS frontend

A165815 Prime congruent numbers (A003273).

5, 7, 13, 23, 29, 31, 37, 41, 47, 53, 61, 71, 79, 101, 103, 109, 127, 137, 149, 151, 157, 167, 173, 181, 191, 197, 199, 223, 229, 239, 257, 263, 269, 271, 277, 293, 311, 313, 317, 349, 353, 359, 367, 373, 383, 389, 397, 421, 431, 439, 457, 461, 463, 479, 487

Offset: 1

T. D. Noe, Sep 28 2009

nonn

Heegner proved that every prime p with p = 5 or 7 (mod 8) is a congruent number. See A003628 for those primes.

Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Zeitschrift 56 (1952), 227-253.

Kent E. Morrison, Congruent Numbers

Cf. A003273, A003628.

A268602 Numerator of the side lengths (legs in ascending order) of the easiest Pythagorean Triangle (with smallest hypotenuse) according to the congruent numbers A003273.

Original entry on oeis.org

3, 20, 41, 3, 4, 5, 35, 24, 337, 780, 323, 106921, 8, 21, 65, 4, 15, 17, 3, 40, 41, 7, 12, 25, 33, 140, 4901, 80155, 41496, 905141617, 6, 8, 10, 35, 48, 337, 99, 52780, 48029801, 5, 12, 13, 720, 8897, 2566561, 17, 24, 145, 450660, 777923, 605170417321, 1700, 5301, 1646021

Offset: 1

Martin Renner, Feb 08 2016

Every three fractions x < y < z satisfy the Pythagorean equation x^2 + y^2 = z^2: (a(3*n-2)/A268603(3*n-2))^2 + (a(3*n-1)/A268603(3*n-1))^2 = (a(3*n)/A268603(3*n))^2.

The area A = x*y/2 of these Pythagorean triangles is a congruent number: A003273(n) = (1/2) * (a(3*n-2)/A268603(3*n-2)) * (a(3*n-1)/A268603(3*n-1)).

			The first congruent number is 5 and the associated right triangle with the side lengths x = 3/2, y = 20/3, z = 41/6 satisfies the Pythagorean equation (3/2)^2 + (20/3)^2 = (41/6)^2 and the area of this triangle equals 1/2*3/2*20/3 = 5.

Editors of OEIS, Discussion of A268602, Apr 22 2023.
Eric Weisstein's World of Mathematics, Congruent Number.

Cf. A003273, A268603.

a(14) corrected on Mar 14 2020

More terms from Jinyuan Wang, Apr 22 2023

A182429 First differences of congruent numbers A003273.

Original entry on oeis.org

1, 1, 6, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 2, 4, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 1, 2, 4, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 6, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1

Offset: 1

N. J. A. Sloane, Apr 28 2012

nonn

Cf. A003273.

A253278 Congruentfree numbers: positive integers that are not divisible by any congruent number (A003273).

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 11, 16, 17, 19, 27, 32, 33, 43, 51, 57, 59, 64, 67, 73, 81, 83, 89, 97, 99, 107, 113, 121, 128, 129, 131, 139, 146, 153, 163, 171, 177, 178, 179, 187, 193, 201, 209, 211, 227, 233, 241, 243, 249, 251, 256, 267, 281, 283, 289, 292, 297, 307

Offset: 1

Juri-Stepan Gerasimov, May 02 2015

nonn

			8 is in this sequence because the positive integer 8 is not divisible by any of the first 3 congruent numbers 5, 6, 7.

Cf. A003273, A257604.

isok(n,v) = {my(k=1); while (v[k] <= n, if (n % v[k] == 0, return (0)); k++;); return (1);}
lista(nn) = {v = readvec("b003273.txt"); for (n=1, nn, if (isok(n, v), print1(n, ", ")););} \\ Michel Marcus, May 03 2015

Corrected by Michel Marcus, May 03 2015. Edited by Wolfdieter Lang, May 05 2015

A268603 Denominator of the side lengths (legs in ascending order) of the easiest Pythagorean Triangle (with smallest hypotenuse) according to the congruent numbers A003273.

Original entry on oeis.org

2, 3, 6, 1, 1, 1, 12, 5, 60, 323, 30, 9690, 3, 2, 6, 1, 2, 2, 1, 3, 3, 2, 1, 2, 35, 3, 105, 20748, 3485, 72306780

Offset: 1

Martin Renner, Feb 08 2016

Every three fractions x < y < z satisfy the Pythagorean equation x^2 + y^2 = z^2: (A268602(3*n-2)/a(3*n-2))^2 + (A268602(3*n-1)/a(3*n-1))^2 = (A268602(3*n)/a(3*n))^2.

The area A = x*y/2 of these Pythagorean triangles is a congruent number: A003273(n) = (1/2) * (A268602(3*n-2)/a(3*n-2)) * (A268602(3*n-1)/a(3*n-1)).

			The first congruent number is 5 and the associated right triangle with the side lengths x = 3/2, y = 20/3, z = 41/6 satisfies the Pythagorean equation (3/2)^2 + (20/3)^2 = (41/6)^2 and the area of this triangle equals 1/2*3/2*20/3 = 5.

Eric Weisstein's World of Mathematics, Congruent Number.

Cf. A003273, A268602.

a(14) corrected on Mar 14 2020

A290349 Least multiplier of n such that n*a(n) becomes a congruent number A003273.

Original entry on oeis.org

5, 3, 2, 5, 1, 1, 1, 3, 5, 2, 2, 2, 1, 1, 1, 5, 2, 3, 2, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 1, 1, 3, 5, 1, 2, 5, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 5, 3, 2, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 1, 1, 5, 1, 3, 2, 2, 1, 1, 1, 3, 3, 2, 2, 2, 1, 1, 1, 1, 5, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Frank M Jackson, Jul 28 2017

nonn

Conditional on the Birch and Swinnerton-Dyer conjecture, it can be shown that the only members of this sequence are the Fibonacci numbers {1,2,3,5}. The underlying pattern of three consecutive 1's per octet shows that numbers congruent to {5,6,7} mod 8 are congruent numbers. Also if n is a square then a(n)=5. This is because all congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a positive square number A000290 and 5 is the least congruent number.

			a(10)=2 as 10*2=20 and 10*3=30 are congruent numbers but 2 is the least multiplier.

Frank M Jackson, Table of n, a(n) for n = 1..100000
Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008.
Giovanni Resta, Congruent numbers Primitive congruent numbers up to 10^7.
Frank M Jackson, Mma program for generating b-file

Cf. A003273, A006991, A273929.

Sfcore[n_] := Module[{m, fac=Select[FactorInteger[n], OddQ[#[[2]]] &]}, If[!SquareFreeQ[n], Times@@Table[fac[[m]][[1]], {m, Length[fac]}], n]]; 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+4y^2+32z^2==n/2, {x, y, z}, Integers], ok=True]; ok)]; lst = {}; Do[AppendTo[lst, (Min[Select[n {1, 2, 3, 5}, CongruentQ[Sfcore[#]] &]])/n], {n, 1, 200}]; lst

A006991 Primitive congruent numbers.

Original entry on oeis.org

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

Offset: 1

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

nonn

Squarefree terms of A003273.

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

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

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 155.
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, 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
Jose Aranda, C++ program
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009.
Clay Mathematics Institute, The Birch and Swinnerton-Dyer 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, and 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), 323-334.
Wikipedia, Congruent number
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

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

(* 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
(* The following self-contained Mathematica code also assumes the truth of the Birch and Swinnerton-Dyer 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 *)

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

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

A248397 Noncongruent squarefree numbers n with A248394(n)/d(n) = 1, where d(n) = A000005(n).

Original entry on oeis.org

1, 3, 33, 51, 57, 59, 83, 139, 177, 187, 209, 211, 267, 321, 339, 345, 379, 385, 411, 451, 489, 499, 515, 555, 587, 595, 649, 659, 665, 681, 707, 803, 811, 827, 835, 899, 921, 969, 1001, 1059, 1099, 1137, 1171, 1211, 1219, 1235, 1259, 1267, 1281, 1315, 1329, 1363

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

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

Cf. A000005, A003273, A165564, A248394, A248395, A248397-A248407.

More terms from Amiram Eldar, Oct 13 2019

A073120 Areas of Pythagorean (or right) triangles with integer sides of the form (2mn, m^2 - n^2, m^2 + n^2).

Original entry on oeis.org

6, 24, 30, 60, 84, 96, 120, 180, 210, 240, 330, 336, 384, 480, 486, 504, 546, 630, 720, 840, 924, 960, 990, 1224, 1320, 1344, 1386, 1536, 1560, 1710, 1716, 1920, 1944, 2016, 2184, 2310, 2340, 2430, 2520, 2574, 2730, 2880, 3036, 3360, 3570, 3696, 3750, 3840

Offset: 1

Zak Seidov, Aug 25 2002

Equivalently, integers of the form m*n*(m^2 - n^2) where m,n are positive integers with m > n. - James R. Buddenhagen, Aug 10 2008
The sequence giving the areas of all Pythagorean triangles is A009112 (sometimes called "Pythagorean numbers").
For example, the sequence does not contain 54, the area of the Pythagorean triangle with sides (9,12,15). - Robert Israel, Apr 03 2015
See also Theorem 2 of Mohanty and Mohanty. - T. D. Noe, Sep 24 2013

			6 = 3*4/2 is the area of the right triangle with sides 3 and 4.
84 = 7*24/2 is the area of the right triangle with sides 7 and 24.

T. D. Noe, Table of n, a(n) for n = 1..10000
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Konstantine Hermes Zelator, A little noticed right triangle, arXiv:0804.1340 [math.GM], 2008.

Cf. A009112, A002144, A003273, A046081, A057102, A024365.

nn = 16; t = Union[Flatten[Table[m*n*(m^2 - n^2), {m, 2, nn}, {n, m - 1}]]]; Select[t, # < nn*(nn^2 - 1) &]

a(n) = A057102(n) / 4. - Max Alekseyev, Nov 14 2008

Description corrected by James R. Buddenhagen, Aug 10 2008, and by Max Alekseyev, Nov 12 2008

Edited by N. J. A. Sloane, Apr 06 2015

cp's OEIS Frontend

A165815 Prime congruent numbers (A003273).

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A268602 Numerator of the side lengths (legs in ascending order) of the easiest Pythagorean Triangle (with smallest hypotenuse) according to the congruent numbers A003273.

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A182429 First differences of congruent numbers A003273.

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A253278 Congruentfree numbers: positive integers that are not divisible by any congruent number (A003273).

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A268603 Denominator of the side lengths (legs in ascending order) of the easiest Pythagorean Triangle (with smallest hypotenuse) according to the congruent numbers A003273.

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Extensions

A290349 Least multiplier of n such that n*a(n) becomes a congruent number A003273.

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Mathematica

A006991 Primitive congruent numbers.

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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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A248397 Noncongruent squarefree numbers n with A248394(n)/d(n) = 1, where d(n) = A000005(n).

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