

A033200


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


9



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
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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 pgon). 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) = (p1)/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 185186
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


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 !! (n1)
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 * A309581 A291277 A191375
Adjacent sequences: A033197 A033198 A033199 * A033201 A033202 A033203


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane.


STATUS

approved



