A141171 Primes of the form -x^2+4*x*y+2*y^2 (as well as of the form 5*x^2+8*x*y+2*y^2).

2, 5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 431, 461, 479, 503, 509, 557, 599, 647, 653, 677, 701, 719, 743, 773, 797, 821, 839, 863, 887, 911, 941, 983, 1013, 1031, 1061, 1103, 1109, 1151, 1181, 1223, 1229, 1277, 1301, 1319, 1367

Offset: 1

Author

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

Keywords

• nonn

Comments

Discriminant is 24. Class is 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 6*u^2 - v^2. The transformation {u, v} = {y, x - 2*y} yields the form in the title. - Juan Arias-de-Reyna, Mar 19 2011

Members of A141171 but not of A105880: 2, 431, 911, 1013, 1181, ..., . - Robert G. Wilson v, Aug 30 2013

This is also the list of primes p such that p = 2 or p is congruent to 5 or 23 mod 24 - Jean-François Alcover, Oct 28 2016

Examples

			a(4) = 29 because we can write 29 = -1^2 + 4*1*3 + 2*3^2 (or 29 = 5*1^2 + 8*1*2 + 2*2^2).
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.
• D. B. Zagier, Zetafunktionen und quadratische Körper.

Links

• 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)

Crossrefs

Cf. A141170 (d = 24), A105880 (Primes for which -8 is a primitive root.) A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).

Cf. also A242665.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

Maple

N:= 10^5: # to get all terms <= N
select(t -> isprime(t) and [isolve(6*u^2-v^2=t)]<>[], [2, seq(op([24*i+5,24*i+23]),i=0..floor((N-5)/24))]); # Robert Israel, Sep 28 2014

Mathematica

A141171 = {}; Do[p = -x^2 + 4 * x * y + 2 * y^2; If[p > 0 && PrimeQ@ p, AppendTo[A141171, p]], {x, 25}, {y, 25}]; Take[ Union@ A141171, 57] (* Robert G. Wilson v, Aug 30 2013 *)
Select[Prime[Range[250]], # == 2 || MatchQ[Mod[#, 24], 5|23]&] (* Jean-François Alcover, Oct 28 2016 *)