A084865 Primes of the form 2x^2 + 3y^2.

2, 3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 251, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 683, 701, 773, 797, 821, 827, 941, 947, 971, 1013, 1019, 1061, 1091, 1109, 1163, 1181, 1187

Offset: 1

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

• nonn
• easy

Comments

Subsequence of A084864; A084863(a(n))>0.

Conjecture: A084863(a(n))=1?

Is it true that a(n) = A019338(n+1)?

Comment: The truth of the conjecture A084863(a(n))=1 follows from the genus theory of quadratic forms (see Cox, page 61). By comparing enough terms, we see that the conjecture a(n) = A019338(n+1) is false. - T. D. Noe, May 02 2008

Appears to be the primes p such that (p mod 6)*(Fibonacci(p) mod 6)=25. - Gary Detlefs, May 26 2014

Examples

			A000040(17) = 59 = 32 + 27 = 2*4^2 + 3*3^2, therefore 59 is a term.
		

References

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

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)

Crossrefs

Cf. A084863, A084864, A019338, A084866, A139827.

Primes in A002480.

Programs

Mathematica

QuadPrimes2[2, 0, 3, 10000] (* see A106856 *)

PARI

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\2), w=2*x^2; for(y=0, sqrtint((lim-w)\3), if(isprime(t=w+3*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

Formula

The primes are congruent to {2, 3, 5, 11} (mod 24). - T. D. Noe, May 02 2008