A033205 Primes of form x^2 + 5*y^2.

5, 29, 41, 61, 89, 101, 109, 149, 181, 229, 241, 269, 281, 349, 389, 401, 409, 421, 449, 461, 509, 521, 541, 569, 601, 641, 661, 701, 709, 761, 769, 809, 821, 829, 881, 929, 941, 1009, 1021, 1049, 1061, 1069, 1109, 1129, 1181, 1201, 1229, 1249, 1289, 1301, 1321, 1361, 1381, 1409, 1429, 1481, 1489

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

It is a classical result that p is of the form x^2 + 5y^2 if and only if p = 5 or p == 1 or 9 mod 20 (see Cox, page 33). - N. J. A. Sloane, Sep 20 2012

Except for 5, also primes of the form x^2 + 25y^2. See A140633. - T. D. Noe, May 19 2008

Or, 5 and all primes p that divide Fibonacci((p - 1)/2) = A121568(n). - Alexander Adamchuk, Aug 07 2006

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 2000 terms from Vincenzo Librandi]
• B. W. Brewer, On primes of the form u^2+5v^2, Am. Math. Monthly vol. 17 no 2 (1966) pp 502-509.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Subsequence of A091729.

Primes in A020669 (numbers of form x^2+5y^2). Cf. A121568, A139643, A216815.

Cf. A029718, A106865 (in the same genus).

Programs

Magma

[p: p in PrimesUpTo(2000) | NormEquation(5,p) eq true]; // Bruno Berselli, Jul 03 2016

Mathematica

QuadPrimes2[1, 0, 5, 10000] (* see A106856 *)

PARI

is(n)=my(k=n%20); n==5 || ((k==9 || k==9) && isprime(n)) \\ Charles R Greathouse IV, Feb 09 2017

Formula

A020669 INTERSECT A000040.

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012