1,1

These are the prime elements of A009003.

-1 is a quadratic residue mod a prime p if and only if p is in this sequence.

Sin(a(n)*Pi/2) = 1 with Pi = 3.1415..., see A070750. - Reinhard Zumkeller, May 04 2002

If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2 = p (mod q), x^2 = q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003

Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004

Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.

Also, primes of the form a^k + b^k, k > 1 (cf. A089716). - Amarnath Murthy, Nov 17 2003

The square of a(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = a(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005

Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk, Aug 10 2006

The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi, Mar 28 2008

A079260(a(n)) = 1; complement of A137409. - Reinhard Zumkeller, Oct 11 2008

From Artur Jasinski, Dec 10 2008: (Start)

If we take 4 numbers: 1, A002314(n), A152676(n), A152680(n) then multiplication table modulo a(n) is isomorphic to the Latin square:

   1 2 3 4

   2 4 1 3

   3 1 4 2

   4 3 2 1

and isomorphic to the multiplication table of {1, i, -i, -1} where i is sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i. 1, A002314(n), A152676(n), A152680(n) are subfield of Galois field [a(n)]. (End)

Primes p such that arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes, this one and A002145. - Ctibor O. Zizka, Oct 20 2009

Equivalently, the primes p for which the smallest extension of F_p containing the square roots of unity (necessarily F_p) contains the 4th roots of unity. In this respect, the n = 2 case of a family of sequences: see n=3 (A129805) and n=5 (A172469). - Katherine Stange (stange(AT)pims.math.ca), Feb 03 2010

a(n) = A000290(A002972(n)) + A000290(2*A002973(n)) = A000290(A002331(n+1)) + A000290(A002330(n+1)). - Reinhard Zumkeller, Feb 16 2010

Primes of form A050993(k)/5. - Juri-Stepan Gerasimov, Jul 01 2010

Subsequence of A007969. - Reinhard Zumkeller, Jun 18 2011

A151763(a(n)) = 1.

2 = 1^2 + 1^2 is also a Pythagorean prime. - Daniel Forgues, Oct 27 2012

n^n - 1 is divisible by 4*n + 1 if 4*n + 1 is a prime (See Dickson reference). - Gary Detlefs, May 22 2013

Not only are the squares of these primes the sum of two nonzero squares, but the primes themselves are also. 2 is the only prime equal to the sum of two nonzero squares and whose square is not. 2 is therefore not a Pythagorean prime. - Jean-Christophe Hervé, Nov 10 2013

The decompositions of the prime and its square into two nonzero squares are unique. - Jean-Christophe Hervé, Nov 11 2013. See the Dickson reference, Vol. II, (B) on p. 227. - Wolfdieter Lang, Jan 13 2015

p^e for p prime of the form 4*k+1 and e>=1 is the sum of 2 nonzero squares. - Jon Perry, Nov 23 2014

Primes p such that the area of the isosceles triangle of sides (p, p, q) for some integer q is an integer. - Michel Lagneau, Dec 31 2014

a(n) is also the set of all primes that are the average of two squares. - Richard R. Forberg, Mar 01 2015

Numbers n such that ((n-3)!!)^2 == -1 (mod n). - Thomas Ordowski, Jul 28 2016

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

L. E. Dickson, "History of the Theory of Numbers", Chelsea Publishing Company, 1919, Vol I, page 386

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.

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).

T. D. Noe and Moshe Levin, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Page 870.

P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014;

C. Banderier, Calcul de (-1/p)

J. Butcher, The Quadratic Residue Theorem

R. Chapman, Quadratic reciprocity

J. E. Ewell, A Simple Proof of Fermat's Two-Square Theorem, The American Mathematical Monthly, Vol. 90, No. 9 (Nov., 1983), pp. 635-637.

Bernard Frénicle de Bessy, Traité des triangles rectangles en nombres : dans lequel plusieurs belles propriétés de ces triangles sont démontrées par de nouveaux principes, Michalet, Paris (1676) pp. 0-116; see p. 44, Consequence II.

Bernard Frénicle de Bessy, Méthode pour trouver la solution des problèmes par les exclusions. Abrégé des combinaisons. Des Quarrez magiques, in "Divers ouvrages de mathématiques et de physique, par MM. de l'Académie royale des sciences", (1693) "Troisième exemple", pp. 17-26, see in particular p 25.

A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

D. & C. Hazzlewood, Quadratic Reciprocity

R. C. Laubenbacher & D. J. Pengelley, Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem

R. C. Laubenbacher & D. J. Pengelley, Gauss, Eisenstein and the 'third' proof of the Quadratic Reciprocity Theorem

K. Matthews, Serret's algorithm based Server

D. Shanks, Review of "K. E. Kloss et al., Class number of primes of the form 4n+1", Math. Comp., 23 (1969), 213-214. [Annotated scanned preprint of review]

S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.

Eric Weisstein's World of Mathematics, Wilson's Theorem

Eric Weisstein's World of Mathematics, Pythagorean Triples

Wolfram Research, The Gauss Reciprocity Law

G. Xiao, Two squares

Wikipedia, Quadratic reciprocity

D. Zagier, A One-Sentence Proof That Every Prime p == 1 (mod 4) Is a Sum of Two Squares, Am. Math. Monthly, Vol. 97, No. 2 (Feb 1990), p. 144. [From Wolfdieter Lang, Jan 17 2015 (thanks to Charles Nash)]

Odd primes of form x^2 + y^2, (x=A002331, y=A002330, with x < y) or of form u^2 + 4*v^2, (u = A002972, v = A002973, with u odd). - Lekraj Beedassy, Jul 16 2004

p^2 - 1 = 12*Sum_{i = 0..floor(p/4)} floor(sqrt(i*p)) where p = a(n) = 4n + 1. [Shirali]

a(n) = (A002972(n)^2 + (2*A002973(n))^2, n >= 1. See the Jean-Christophe Hervé Nov 11 2013 comment. - Wolfdieter Lang, Jan 13 2015

The following table shows the relationship between several closely related sequences:

Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;

a = A002331, b = A002330, t_1 = ab/2 = A070151;

p^2 = c^2 + d^2 with c < d; c = A002366, d = A002365,

t_2 = 2ab = A145046, t_3 = b^2 - a^2 = A070079,

with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).

---------------------------------

.p..a..b..t_1..c...d.t_2.t_3..t_4

---------------------------------

.5..1..2...1...3...4...4...3....6

13..2..3...3...5..12..12...5...30

17..1..4...2...8..15...8..15...60

29..2..5...5..20..21..20..21..210

37..1..6...3..12..35..12..35..210

41..4..5..10...9..40..40...9..180

53..2..7...7..28..45..28..45..630

.................................

a(7) = 53 = A002972(7)^2 + (2*A002973(n))^2 = 7^2 + (2*1)^2 = 49 + 4, and this is the only way. - Wolfdieter Lang, Jan 13 2015

a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a), 4*n+1]; fi; od: A002144 := n->a[n];

# 2nd program

A002144 := proc(n)

    option remember;

    local a;

    if n = 1 then

        5;

    else

        for a from procname(n-1)+4 by 4 do

            if isprime(a) then

                return a;

            end if;

        end do:

    end if;

end proc: # R. J. Mathar, May 26 2016

Select[4*Range[140] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 16 2006 *)

pythPrimes = {}; Do[If[Mod[Prime[n], 4] == 1, AppendTo[pythPrimes, Prime[n]]], {n, 1, 200}]; pythPrimes (* Artur Jasinski, Dec 10 2008 *)

Select[Range[5, 617, 4], PrimeQ] (* Zak Seidov, Aug 31 2012 *)

Select[ Prime@ Range[2, 110], Length@ PowersRepresentations[#^2, 2, 2] > 1 &] (* or *)

Select[ Prime@ Range[2, 110], JacobiSymbol[-1, #] == 1 &] (* Robert G. Wilson v, May 11 2014 *)

nn=1000; lst={}; Do[p=Prime[a]; s=p^2-b^2/4; If[0<s&&IntegerQ[(b/2)*Sqrt[s]], AppendTo[lst, p]], {a, nn}, {b, nn}]; Union[lst] (* Michel Lagneau, Dec 31 2014 *)

(PARI) select(primes(1000), p->p%4==1) \\ version 2.4.2 or before

(PARI) select(p->p%4==1, primes(1000)) \\ newer versions

(Haskell)

a002144 n = a002144_list !! (n-1)

a002144_list = filter ((== 1) . a010051) [1, 5..]

-- Reinhard Zumkeller, Mar 06 2012, Feb 22 2011

(PARI) {a(n) = local(m, c); if( n<1, 0, c = 0; m = 0; while( c<n, m++; if( m%4 == 1 && isprime(m), c++)); m)} /* Michael Somos, Mar 10 2012 */

(Sage)

def A002144_list(n): # returns all Pythagorean primes <= n

    return filter(is_prime, range(5, n+1, 4))

A002144_list(617) # Peter Luschny, Sep 12 2012

(Python)

from sympy import prime

A002144 = [n for n in (prime(x) for x in range(1, 10**3)) if not (n-1) % 4]

# Chai Wah Wu, Sep 01 2014

(MAGMA) [a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014

For values of n see A005098. Cf. A002145, A002476. Apart from initial term, same as A002313. Cf. A114200, A003658, A002314, A152676, A152680, A173330, A173331, A010051; A007519, A094407, A133870, A142925, A208177, A208178, A076339.

Cf. A004613 (multiplicative closure).

Primes in A020668.

Cf. A002972, A002973.

Sequence in context: A231754 A175768 * A280084 A192592 A111055 A283391

Adjacent sequences:  A002141 A002142 A002143 * A002145 A002146 A002147

nonn,easy,nice

N. J. A. Sloane, Apr 30 1991

approved