A001132 Primes == +-1 (mod 8).

7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Primes p such that 2 is a quadratic residue mod p.

Also primes p such that p divides 2^((p-1)/2) - 1. - Cino Hilliard, Sep 04 2004

A001132 is exactly formed by the prime numbers of A118905: in fact at first every prime p of A118905 is p = u^2 - v^2 + 2uv, with for example u odd and v even so that p - 1 = 4u'(u' + 1) + 4v'(2u' + 1 - v') when u = 2u' + 1 and v = 2v'. u'(u' + 1) is even and v'(2u' + 1 - v') is always even. At second hand if p = 8k +- 1, p has the shape x^2 - 2y^2; letting u = x - y and v = y, comes p = (x - y)^2 - y^2 + 2(x - y)y = u^2 - v^2 + 2uv so p is a sum of the two legs of a Pythagorean triangle. - Richard Choulet, Dec 16 2008

These are also the primes of form x^2 - 2y^2, excluding 2. See A038873. - Tito Piezas III, Dec 28 2008

Primes p such that p^2 mod 48 = 1. - Gary Detlefs, Dec 29 2011

Primes in A047522. - Reinhard Zumkeller, Jan 07 2012

This sequence gives the odd primes p which satisfy C(p, x = 0) = +1, where C(p, x) is the minimal polynomial of 2*cos(Pi/p) (see A187360). For the proof see a comment on C(n, 0) in A230075. - Wolfdieter Lang, Oct 24 2013

Each a(n) corresponds to precisely one primitive Pythagorean triangle. For a proof see the W. Lang link, also for a table. See also the comment by Richard Choulet above, where the case u even and v odd has not been considered. - Wolfdieter Lang, Feb 17 2015

Primes p such that p^2 mod 16 = 1. - Vincenzo Librandi, May 23 2016

Rational primes that decompose in the field Q(sqrt(2)). - N. J. A. Sloane, Dec 26 2017

References

• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
• Ronald S. Irving, Integers, Polynomials, and Rings. New York: Springer-Verlag (2004): 274.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• C. Banderier, Calcul de (2/p)
• Wolfdieter Lang, A001132: Primes +1 (mod 8) or -1 (mod 8) and Sum of Legs of Primitive Pythagorean Triangles.
• Index entries for related sequences
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

For primes p such that x^m = 2 (mod p) has a solution see A001132 (for m = 2), A040028 (m = 3), A040098 (m = 4), A040159 (m = 5), A040992 (m = 6), A042966 (m = 7), A045315 (m = 8), A049596 (m = 9), A049542 (m = 10) - A049595 (m = 63). Jeff Lagarias (lagarias(AT)umich.edu) points out that all these sequences are different, although this may not be apparent from looking just at the initial terms.

Agrees with A038873 except for initial term.

Cf. A118905, A010051.

Union of A007519 and A007522.

Programs

Haskell

a001132 n = a001132_list !! (n-1)
a001132_list = [x | x <- a047522_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

Magma

[p: p in PrimesUpTo (600) | p^2 mod 16 eq 1]; // Vincenzo Librandi, May 23 2016

Maple

seq(`if`(member(ithprime(n) mod 8, {1,7}),ithprime(n),NULL),n=1..109); # Nathaniel Johnston, Jun 26 2011
for n from 1 to 600 do if (ithprime(n)^2 mod 48 = 1) then print(ithprime(n)) fi od. # Gary Detlefs, Dec 29 2011

Mathematica

Select[Prime[Range[250]], MemberQ[{1, 7}, Mod[#, 8]] &]  (* Harvey P. Dale, Apr 29 2011 *)
Select[Union[8Range[100] - 1, 8Range[100] + 1], PrimeQ] (* Alonso del Arte, May 22 2016 *)

PARI

select(p->p%8==1 ||p%8==7, primes(100)) \\ Charles R Greathouse IV, May 18 2015

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, May 18 2015