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

1,1

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

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

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

Example

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.

Mathematica

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

Prog

(Magma) [p: p in PrimesUpTo(600) | p mod 8 in [1, 3]]; // Vincenzo Librandi, Aug 04 2012
(Haskell)
a033200 n = a033200_list !! (n-1)
a033200_list = filter ((== 1) . a010051) a047471_list
-- Reinhard Zumkeller, Dec 29 2012
(PARI) is(n)=n%8<4 && n%2 && isprime(n) \\ Charles R Greathouse IV, Feb 09 2017

Crossrefs

Cf. A033203.

Sequence in context: A095280 A085317 A210311 * A369171 A309581 A291277

Adjacent sequences: A033197 A033198 A033199 * A033201 A033202 A033203

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved