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
A106865 Primes of the form 2x^2 + 2xy + 3y^2.
2, 3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187
Offset: 1
Comments
Discriminant = -20.
Also: Primes of the form 2x^2 - 2xy + 3y^2 with x and y nonnegative. Cf. A106864.
Primes congruent to 2, 3, 7 modulo 20. - Michael Somos, Aug 13 2006
In Z[sqrt(-5)], these numbers are irreducible but not prime. In terms of ideals, they generate principal ideals that are not prime (or maximal). The equation x^2 + 5y^2 = a(n) has no solutions, but x^2 = -5 (mod a(n)) does. For example, 2 * 3 = (1 - sqrt(-5))(1 + sqrt(-5)) and 7 * 23 = (9 - 4*sqrt(-5))(9 + 4*sqrt(-5)). - Alonso del Arte, Dec 19 2015
Examples
x = 1, y = 1 gives 2x^2 + 2xy + 3y^2 = 2 + 2 + 3 = 7. x = 1, y = -3 gives 2x^2 + 2xy + 3y^2 = 2 - 6 + 27 = 23.
References
- David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.
Links
- Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Programs
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Maple
select(isprime, [2, seq(seq(5+s+20*i,s=[-2,2]),i=0..10^3)]); # Robert Israel, Dec 23 2015
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Mathematica
QuadPrimes2[2, -2, 3, 10000] (* see A106856 *)
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PARI
is(n)=isprime(n) && #qfbsolve(Qfb(2,2,3),n)>0 \\ Charles R Greathouse IV, Feb 09 2017
A242655 Numbers of the form 2x^2+2xy+3y^2 with x and y nonnegative.
0, 2, 3, 7, 8, 12, 15, 18, 27, 28, 32, 35, 42, 43, 47, 48, 50, 58, 60, 63, 72, 75, 82, 83, 87, 90, 98, 103, 107, 108, 112, 115, 122, 123, 128, 135, 138, 140, 147, 162, 163, 167, 168, 172, 175, 183, 188, 192, 200, 202, 203, 207, 210, 218, 223, 232, 235, 240, 242, 243, 252, 258, 263, 267, 282, 283, 287, 288, 290, 298, 300
Offset: 1
Keywords
Comments
Discriminant = -20.
Links
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Comments
References
Links
Crossrefs
Programs
Mathematica
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 *)PARI
Extensions