A106950 -id:A106950 - cp's OEIS frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821

Offset: 1

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
QuadPrimes2[1, 1, 2, 1000]
(This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014)
max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

A141170 Primes of the form x^2+4xy-2y^2 (as well as of the form 3x^2+6xy+y^2).

Original entry on oeis.org

3, 19, 43, 67, 73, 97, 139, 163, 193, 211, 241, 283, 307, 313, 331, 337, 379, 409, 433, 457, 499, 523, 547, 571, 577, 601, 619, 643, 673, 691, 739, 769, 787, 811, 859, 883, 907, 937, 1009, 1033, 1051, 1123, 1129, 1153, 1171, 1201, 1249, 1291, 1297, 1321, 1459, 1483, 1489, 1531

Offset: 1

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

nonn

Discriminant = 24. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
Also, primes of form u^2 - 6v^2. The transformation {u,v} = {x+2y,y} yields the form in the title. - Tito Piezas III, Dec 31 2008
Conjecture: this is also the list of primes that are simultaneously of the form x^2+2y^2 and of the form x^2+3y^2; that is, the intersection of A002476 and A033203. - Zak Seidov, Jun 07 2014
This is also the list of primes p such that p = 3 or p is congruent to 1 or 19 mod 24. - Jean-François Alcover, Oct 28 2016

			a(2)=19 because we can write 19=3^2+4*3*1-2*1^2 (or 19=3*1^2+6*1*2+2^2)

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. A141171 (d=24), A106950 (Primes of the form x^2+18y^2), A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Cf. also A242661, A002476, A033203.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

xy[{x_, y_}]:={x^2 + 4 x y - 2 y^2, y^2 + 4 x y - 2 x^2}; Union[Select[Flatten[xy/@Subsets[Range[40], {2}]], #>0&&PrimeQ[#]&]] (* Vincenzo Librandi, Jun 09 2014 *)
Select[Prime[Range[250]], # == 3 || MatchQ[Mod[#, 24], 1|19]&] (* Jean-François Alcover, Oct 28 2016 *)

A139642 Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n) is a convenient number.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 7, 1, 5, 9, 13, 1, 5, 9, 1, 7, 1, 7, 9, 11, 15, 23, 25, 1, 9, 17, 25, 1, 13, 25, 1, 9, 11, 19, 1, 13, 25, 37, 1, 9, 13, 17, 25, 29, 49, 1, 19, 31, 49, 1, 9, 17, 25, 33, 41, 49, 57, 1, 19, 25, 43, 49, 67, 1, 25, 37, 1, 9, 15, 23, 25, 31, 47, 49, 71, 81, 1, 25, 49, 73, 1, 9

Offset: 1

T. D. Noe, Apr 28 2008

Each row begins with 1. For example, the 12th row is for N=13. The numbers in that row are 1, 9, 17, 25, 29 and 49, which means that the primes represented by the quadratic form x^2+13y^2 (A033210) are congruent to 1, 9, 17, 25, 29,or 49 (mod 52). Cox lists some of these congruences on page 36 of his book. As mentioned by Cox, for these N, every term of the congruence has the form b^2 or N+b^2 for some integer b. In some cases, the congruences can be simplified. For instance, for N=18 (A106950), the congruence is 1, 19, 25, 43, 49, 67 (mod 72), which can be simplified to 1, 19 (mod 24).

			1, 2,
1, 2, 3,
1, 3, 7,
1, 5, 9, 13,
1, 5, 9,
1, 7,
1, 7, 9, 11, 15, 23, 25,
1, 9, 17, 25,
1, 13, 25,
1, 9, 11, 19,
1, 13, 25, 37,
1, 9, 13, 17, 25, 29, 49,
1, 19, 31, 49,
1, 9, 17, 25, 33, 41, 49, 57,
1, 19, 25, 43, 49, 67,
1, 25, 37,
1, 9, 15, 23, 25, 31, 47, 49, 71, 81,
1, 25, 49, 73,
...

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Section 3.

T. D. Noe, Rows n=1..65 of triangle, flattened
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

See the Binary Quadratic Forms and OEIS link for full list of primes generated by x^2+Ny^2, where N is a convenient number.

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cp's OEIS Frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

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A141170 Primes of the form x^2+4xy-2y^2 (as well as of the form 3x^2+6xy+y^2).

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A139642 Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n) is a convenient number.

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cp's OEIS Frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A141170 Primes of the form x^2+4*x*y-2*y^2 (as well as of the form 3*x^2+6*x*y+y^2).

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Mathematica

A139642 Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n) is a convenient number.

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Author

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Comments

Examples

References

Links

Crossrefs

A141170 Primes of the form x^2+4xy-2y^2 (as well as of the form 3x^2+6xy+y^2).