A242662 -id:A242662 - cp's OEIS frontend

A141172 Primes of the form 2x^2+2x*y-3y^2 (as well as of the form 2x^2+6xy+y^2).

2, 29, 37, 53, 109, 113, 137, 149, 193, 197, 233, 277, 281, 317, 337, 373, 389, 401, 421, 449, 457, 541, 557, 569, 613, 617, 641, 653, 673, 701, 709, 757, 809, 821, 877, 953, 977, 1009, 1033, 1061, 1093, 1117, 1129, 1201, 1213, 1229, 1289, 1297, 1373, 1381, 1409, 1429, 1453, 1481, 1493

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008

nonn

Discriminant = 28. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.
Also, primes of form u^2-7v^2. The transformation {u,v}={3x+y,x} yields the second quadratic form given in the title. - Tito Piezas III, Dec 28 2008
This is also the list of primes p such that p = 2 or p is congruent to 1, 9 or 25 mod 28 - Jean-François Alcover, Oct 28 2016

			a(2)=29 because we can write 29=2*4^2+2*4*3-3*3^2 (or 29=2*1^2+6*1*3+3^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A141173 (d=28) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Cf. also A242662.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[Prime[Range[250]], # == 2 || MatchQ[Mod[#, 28], 1|9|25]&] (* Jean-François Alcover, Oct 28 2016 *)

A358946 Positive integers that are properly represented by each primitive binary quadratic form of discriminant 28 that is properly equivalent to the principal form [1, 4, -3].

Original entry on oeis.org

1, 2, 9, 18, 21, 29, 37, 42, 53, 57, 58, 74, 81, 93, 106, 109, 113, 114, 133, 137, 141, 149, 162, 177, 186, 189, 193, 197, 217, 218, 226, 233, 249, 261, 266, 274, 277, 281, 282, 298, 309, 317, 329, 333, 337, 354, 361, 373, 378, 386, 389, 393, 394, 401, 413, 417, 421, 434, 449, 457, 466, 477, 498, 501

Offset: 1

Wolfdieter Lang, Jan 10 2023

nonn

This is a subsequence of A242662, excluding the primitive forms of discriminant 28 with only improper representations of k, like k = 4, 8, 16, 25, 32, ... .
An indefinite binary quadratic primitive form F = a*x^2 + b*x*y + c*y^2 (gcd(a, b, c) = 1) with discriminant Disc = b^2 - 4*a*c = 28 = 2^2*7 is denoted by [a, b, c], or in matrix notation by MF = Matrix([[a, b/2], [b/2, c]]). Hence F = X*MF*X^T (T for transposed), where X = (x, y). See the two links for details and references.
Properly equivalent forms F' and F are related by a matrix R of determinant +1 like MF' = R^T*MF*R, and X'^T = R^{-1}*X^T.
Each primitive form, properly equivalent to the reduced principal form F_p = [1, 4, -3] for Disc = 28 (used in A242662), represents the given nonnegative k = a(n) values (and only these) properly with X = (x, y) and gcd(x, y) = 1. Modulo an overall sign change in X one can choose x nonnegative.
There are 8 = A082174(8) primitive reduced forms of Disc = 28 leading to 2 = A087048(8) (class number) cycles each of period 4, namely the principal cycle CyP = [[1, 4, -3], [-3, 2, 2], [2, 2, -3], [-3, 4, 1]] and the one (with outer signs flipped) CyP' = [[-1, 4, 3], [3, 2, -2], [-2, 2, 3], [3, 4, -1]].
There are A358947(n) representative parallel primitive forms (rpapfs) of discriminant Disc = 28 for k = a(n). This gives the number of proper fundamental representations X = (x, y), with x >= 0, of each primitive form [a, b, c], properly equivalent to the principal form F_p of Disc = 28.
For the negative integers k properly represented by primitive forms [a, b, c] properly equivalent to the principal form of Disc = 28 see A359476. The corresponding number of fundamental proper representations is given in A359477.
This and the three related sequences originated from a proposal by Klaus Purath proving that the form FKP := [1, -2, -6] of Disc = 28 represents k = k(m) = m^2 - 7 = A028881(m), for m >= 3, with the two fundamental representations X1(m) = (m+1, 1) and X2(m) = (11*m - 29, 3*m - 8). This form FKP is properly equivalent to the principal form F_p with R = Matrix([[1, -3], [0, 1]]). Hence all k = a(n) are represented by the form FKP, and A028881 is a subsequence of the present one.

			k = 9 = a(3): F = FPell = [1, 0, -7] is properly equivalent to F_p = [1, 4, -3] by two so-called half-reduced right neighbor R(t)-transformations, with the matrix R = R(t) = Matrix([[0, -1], [1, t]]), first with t = 0 then with t = 2. For FPell representing k = 9 with x > 0 and y > 0 see X_1(9, i) = (A307168(i), A307169(i)) and X_2(9, i) = (A307172(i), A307173(i)), for i >= 0. There are also the representations with y -> -y arising from the opposite fundamental solutions.
The 2 = A358947(3) rpapfs are F1 = [9, 8, 1] and F2 = [9, 10, 2]. They lead by proper equivalence transformations to a form of the above given principal cycle CyP. F1 -> [1, 4, -3] = F_p with matrix R(6), and F2 -> [2, 2, -3] with R(3). See the FIGURE, p. 10, of the linked paper.
Besides the primitive forms FPell, F1, F2 and the four forms of CyP also F' = [-7, 0, 1], and all primitive and properly equivalent forms represent k = 9. See the mentioned FIGURE, where FPa1 = F1, FPa1 = F2, Fpa2' = F_p^{(2)} = [2, 2, -3] and FPa2'' = F_p^{(3)} = [-3, 4, 1].

Wolfdieter Lang, Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/-4 Equations.
Wolfdieter Lang, Cycles of reduced Pell forms, general Pell equations and Pell graphs

Cf. A028881, A242662, A307168, A307169, A307172, A307173, A358947, A359476, A359477.

A242666 Nonnegative integers of the form -x^2 + 4xy + 3y^2.

Original entry on oeis.org

0, 3, 6, 7, 12, 14, 19, 24, 27, 28, 31, 38, 47, 48, 54, 56, 59, 62, 63, 75, 76, 83, 87, 94, 96, 103, 108, 111, 112, 118, 124, 126, 131, 139, 147, 150, 152, 159, 166, 167, 171, 174, 175, 188, 192, 199, 203, 206, 216, 222, 223, 224, 227, 236, 243, 248, 251, 252, 259, 262, 271, 278, 279, 283, 294, 300

Offset: 1

N. J. A. Sloane, May 31 2014

nonn

Discriminant 28.
Nonnegative numbers of the form 7x^2 - y^2. - Jon E. Schoenfield, Jun 03 2022
For the subsequence of the numbers with proper representations (gcd(x, y) = 1) see A359476. ~ Wolfdieter Lang, Jan 17 2023

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in this sequence = A141173.

Cf. A242662, A359476.

Reap[For[n = 0, n <= 300, n++, If[Reduce[-x^2 + 4*x*y + 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

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A141172 Primes of the form 2x^2+2x*y-3y^2 (as well as of the form 2x^2+6xy+y^2).

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A358946 Positive integers that are properly represented by each primitive binary quadratic form of discriminant 28 that is properly equivalent to the principal form [1, 4, -3].

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A242666 Nonnegative integers of the form -x^2 + 4xy + 3y^2.

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Author

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Mathematica

cp's OEIS Frontend

A141172 Primes of the form 2*x^2+2*x*y-3*y^2 (as well as of the form 2*x^2+6*x*y+y^2).

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A358946 Positive integers that are properly represented by each primitive binary quadratic form of discriminant 28 that is properly equivalent to the principal form [1, 4, -3].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A242666 Nonnegative integers of the form -x^2 + 4xy + 3y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A141172 Primes of the form 2x^2+2x*y-3y^2 (as well as of the form 2x^2+6xy+y^2).