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Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.
15

%I #74 Jul 24 2024 12:16:36

%S 3,11,17,19,41,43,59,67,73,83,89,97,107,113,131,137,139,163,179,193,

%T 211,227,233,241,251,257,281,283,307,313,331,337,347,353,379,401,409,

%U 419,433,443,449,457,467,491,499

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

%C Rational primes that decompose in the field Q(sqrt(-2)). - _N. J. A. Sloane_, Dec 25 2017

%C 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

%C Terms m in A047471 with A010051(m) = 1. - _Reinhard Zumkeller_, Dec 29 2012

%C 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

%C 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

%C 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

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

%H Ray Chandler, <a href="/A033200/b033200.txt">Table of n, a(n) for n = 1..10000</a> [First 1000 terms from T. D. Noe]

%H G. Frei, <a href="http://dx.doi.org/10.1007/BF03025809">Euler's convenient numbers</a>, Math. Intell. Vol. 7 No. 3 (1985), p. 56.

%H Zak Seidov, <a href="/A033200/a033200.dat.txt">Table of n, a(n), x and y for n = 1..1000</a>

%H 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)

%H Paul Yiu, <a href="https://cms.math.ca/crux/v24/n3/page175-177.pdf">CRUX, Problem 2331, Proposed by Paul Yiu</a>

%H Paul Yiu and Jill S. Taylor, <a href="http://dongthapvietnam.informe.com/blog/uploads/2012/10/cruxv25_1999.pdf">CRUX, Problem 2331, Solution</a> pp 185-186

%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>

%F a(n) = A033203(n+1). - _Zak Seidov_, May 29 2014

%F A007519 UNION A007520. - _R. J. Mathar_, Jun 09 2020

%F L(-2, a(n)) = +1, n >= 1, with the Legendre symbol L. -_Wolfdieter Lang_, Jul 24 2024

%e 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)).)

%e Since 17 is prime and 17 == 1 (mod 8), 17 is in the sequence.

%t Rest[QuadPrimes2[1, 0, 2, 10000]] (* see A106856 *)

%t Select[Prime[Range[200]],MemberQ[{1,3},Mod[#,8]]&] (* _Harvey P. Dale_, Jun 09 2017 *)

%o (Magma) [p: p in PrimesUpTo(600) | p mod 8 in [1, 3]]; // _Vincenzo Librandi_, Aug 04 2012

%o (Haskell)

%o a033200 n = a033200_list !! (n-1)

%o a033200_list = filter ((== 1) . a010051) a047471_list

%o -- _Reinhard Zumkeller_, Dec 29 2012

%o (PARI) is(n)=n%8<4 && n%2 && isprime(n) \\ _Charles R Greathouse IV_, Feb 09 2017

%Y Cf. A033203.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_