A106908 Primes of the form 4x^2+3xy+4y^2, with x and y any integer.

5, 11, 31, 89, 179, 181, 191, 331, 401, 419, 421, 449, 521, 599, 619, 641, 709, 719, 829, 859, 911, 971, 991, 1021, 1039, 1109, 1259, 1291, 1301, 1489, 1511, 1549, 1621, 1709, 1831, 1871, 1879, 2011, 2039, 2099, 2161, 2179, 2281, 2311, 2381

Offset

1,1

Comments

Discriminant=-55.

Primes p such that x^4 - 2x^3 + x - 1 has no integer roots modulo p^2. According to the Bunyakovsky conjecture, there are infinitely many such primes. - Griffin N. Macris, Jul 21 2016

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)

Wikipedia, Bunyakovsky conjecture

Example

89 is in the sequence since it is prime and with x=5 and y=-1, 4x^2 + 3xy + 4y^2 = 4*5^2 + 3*5*(-1) + 4*(-1)^2 = 100 - 15 + 4 = 89. - Michael B. Porter, Jul 22 2016

Maple

N:= 10000: # to get all terms <= N
S:= {}:
for y from 0 to floor(sqrt(16*N/55)) do
   s:= floor(sqrt(16*N-55*y^2));
   S:= S union select(isprime, {seq(4*x^2 + 3*x*y + 4*y^2,
      x = ceil((-s-3*y)/8) .. floor((s-3*y)/8))})
od:
sort(convert(S, list)); # Robert Israel, Jul 21 2016

Mathematica

Union[QuadPrimes2[4, 3, 4, 10000], QuadPrimes2[4, -3, 4, 10000]] (* see A106856 *)

Crossrefs

Sequence in context: A023276 A074648 A236428 * A107442 A349611 A323867

Adjacent sequences: A106905 A106906 A106907 * A106909 A106910 A106911

Keyword

nonn,easy

Author

T. D. Noe, May 09 2005

Status

approved