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A002144 Pythagorean primes: primes of form 4n + 1.
(Formerly M3823 N1566)
254
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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+4), A002366(n+4), a(n)}.

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

The square of A002144(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = A002144(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

Bernard Frénicle de Bessy demonstrated several times that primes of the form 4*n + 1 were the only ones that could be the hypothenuse of a primitive integer right triangle. In the original "Il s'ensuit aussi que l'hypotenuse d'un triangle primitif surpasse de l'unité un quaternaire." see reference "Traité des triangles ..." He also demonstrated that such an hypothenuse must be the sum of two squares. - Olivier Gérard, Jul 12 2014 after Paul Curtz, Sep 05 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 A002144(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 [A002144(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]

The Pythagorean triple is {A002365(n), A002366(n), a(n)}. - Thomas M. Green, Jun 01 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

REFERENCES

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

Bernard Frénicle de Bessy (1605?-1675), "Traité des triangles rectangles en nombres : dans lequel plusieurs belles propriétés de ces triangles sont démontrées par de nouveaux principes", Michalet, Paris, 1676; see p. 44, Consequence II.

Bernard Frénicle de Bessy (1605?-1675), "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", p17-26, see in particular p25.

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).

LINKS

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

Bernard Frénicle de Bessy, Traité des triangles rectangles en nombres [...] B.N.F. permanent link to a scan of an original print

A. Granville and G. Martin, Prime number races

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

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

FORMULA

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].

EXAMPLE

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

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

MAPLE

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];

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

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.

Sequence in context: A077426 A231754 A175768 * A192592 A111055 A145016

Adjacent sequences:  A002141 A002142 A002143 * A002145 A002146 A002147

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Apr 21 2000

STATUS

approved

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Last modified October 24 03:14 EDT 2014. Contains 248491 sequences.