A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Rational primes that decompose in the field Q(sqrt(-2)). - N. J. A. Sloane, Dec 25 2017

Fermat knew of the relationship between a prime being congruent to 1 or 3 mod 8 and its being the sum of a square and twice a square, and claimed to have a firm proof of this fact. These numbers are not primes in Z[sqrt(-2)], as they have x - y sqrt(-2) as a divisor. - Alonso del Arte, Dec 07 2012

Terms m in A047471 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012

This sequence gives the primes p which satisfy norm(rho(p)) = + 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For the minimal polynomial C see A187360. For p == 1 (mod 8) the norm is C(p, 0) (see a comment on 4*A005123) and for p == 3 (mod 8) the norm is -C(p, 0) (see a comment on A186297). For the primes with norm(rho(p)) = -1 see A003628. - Wolfdieter Lang, Oct 24 2013

If p is a member then it has a unique representation as x^2+2y^2 [Frei, Theorem 3]. - N. J. A. Sloane, May 30 2014

Primes that are the quarter perimeter of a Heronian triangle. Such primes are unique to the Heronian triangle (see Yiu link). - Frank M Jackson, Nov 30 2014

Examples

			Since 11 is prime and 11 == 3 (mod 8), 11 is in the sequence. (Also 11 = 3^2 + 2 * 1^2 = (3 + sqrt(-2))(3 - sqrt(-2)).)
Since 17 is prime and 17 == 1 (mod 8), 17 is in the sequence.
		

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from T. D. Noe]
• G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), p. 56.
• Zak Seidov, Table of n, a(n), x and y for n = 1..1000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• Paul Yiu, CRUX, Problem 2331, Proposed by Paul Yiu
• Paul Yiu and Jill S. Taylor, CRUX, Problem 2331, Solution pp 185-186
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

Cf. A033203.

Programs

Haskell

a033200 n = a033200_list !! (n-1)
a033200_list = filter ((== 1) . a010051) a047471_list
-- Reinhard Zumkeller, Dec 29 2012

Magma

[p: p in PrimesUpTo(600) | p mod 8 in [1, 3]]; // Vincenzo Librandi, Aug 04 2012

Mathematica

Rest[QuadPrimes2[1, 0, 2, 10000]] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,3},Mod[#,8]]&] (* Harvey P. Dale, Jun 09 2017 *)

PARI

is(n)=n%8<4 && n%2 && isprime(n) \\ Charles R Greathouse IV, Feb 09 2017

Formula

a(n) = A033203(n+1). - Zak Seidov, May 29 2014

A007519 UNION A007520. - R. J. Mathar, Jun 09 2020

L(-2, a(n)) = +1, n >= 1, with the Legendre symbol L. -Wolfdieter Lang, Jul 24 2024