

A033200


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


11



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
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



FORMULA



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


CROSSREFS



KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



