cp's OEIS Frontend

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

%I A033205 #54 Jul 08 2025 19:43:57
%S A033205 5,29,41,61,89,101,109,149,181,229,241,269,281,349,389,401,409,421,
%T A033205 449,461,509,521,541,569,601,641,661,701,709,761,769,809,821,829,881,
%U A033205 929,941,1009,1021,1049,1061,1069,1109,1129,1181,1201,1229,1249,1289,1301,1321,1361,1381,1409,1429,1481,1489
%N A033205 Primes of form x^2 + 5*y^2.
%C A033205 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
%C A033205 Except for 5, also primes of the form x^2 + 25y^2. See A140633. - _T. D. Noe_, May 19 2008
%C A033205 Or, 5 and all primes p that divide Fibonacci((p - 1)/2) = A121568(n). - _Alexander Adamchuk_, Aug 07 2006
%D A033205 David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.
%H A033205 Vincenzo Librandi and Ray Chandler, <a href="/A033205/b033205.txt">Table of n, a(n) for n = 1..10000</a> [First 2000 terms from Vincenzo Librandi]
%H A033205 B. W. Brewer, <a href="http://www.jstor.org/stable/2035200">On primes of the form u^2+5v^2</a>, Am. Math. Monthly vol. 17 no 2 (1966) pp 502-509.
%H A033205 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%F A033205 A020669 INTERSECT A000040.
%F A033205 a(n) ~ 4n log n. - _Charles R Greathouse IV_, Nov 09 2012
%t A033205 QuadPrimes2[1, 0, 5, 10000] (* see A106856 *)
%o A033205 (Magma) [p: p in PrimesUpTo(2000) | NormEquation(5,p) eq true]; // _Bruno Berselli_, Jul 03 2016
%o A033205 (PARI) is(n)=my(k=n%20); n==5 || ((k==9 || k==9) && isprime(n)) \\ _Charles R Greathouse IV_, Feb 09 2017
%Y A033205 Subsequence of A091729.
%Y A033205 Primes in A020669 (numbers of form x^2+5y^2). Cf. A121568, A139643, A216815.
%Y A033205 Cf. A029718, A106865 (in the same genus).
%K A033205 nonn,easy
%O A033205 1,1
%A A033205 _N. J. A. Sloane_