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A002144
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Pythagorean primes: primes of form 4n+1.
(Formerly M3823 N1566)
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169
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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; internal format)
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OFFSET
| 1,1
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COMMENTS
| These are the primitive elements of A009003.
-1 is a quadratic residue mod a prime p iff p is in this sequence.
sin(a(n)*pi/2) = 1 with pi=3.1415..., see A070750. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (blekraj(AT)yahoo.com), Jul 17 2003
Odd primes such that binomial(p-1,(p-1)/2) == 1 (mod p) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (amarnath_murthy(AT)yahoo.com), 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 (alex(AT)kolmogorov.com), Aug 10 2006
The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi (paulmuljadi(AT)yahoo.com), Mar 28 2008
Frenicle mentionned 4n+1 for primes : Methode pour trouver .., page 14 on 44.In Divers ouvrages de mathematique .. .In-folio,6,518,1 pp,Paris,1693. [From Paul Curtz (bpcrtz(AT)free.fr), Sep 05 2008]
A079260(a(n)) = 1; complement of A137409. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 11 2008]
Contribution from Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008: (Start)
If we take 4 numbers : 1, A002314(n), A152676(n), A152680(n) then
multiplication table modulo A002144(n) is isomorphc 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. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Oct 20 2009]
Contribution from Katherine Stange (stange(AT)pims.math.ca), Feb 03 2010: (Start)
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). (End)
a(n) = A000290(A002972(n))+A000290(2*A002973(n)) = A000290(A002331(n+1))+A000290(A002330(n+1)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 16 2010]
The Pythagorean triple is {A002365(n), A002366(n),a(n)}. [From Thomas M. Green (tgreen(AT)astound.net), Jun 01 2010]
Primes of form n-th 5-Knodel number/5. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jul 01 2010]
Subsequence of A007969. [Reinhard Zumkeller, Jun 18 2011]
A151763(a(n)) = 1.
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REFERENCES
| D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. 70 (4) (1997) 263.
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).
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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.
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
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 Quaratic 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
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
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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 (blekraj(AT)yahoo.com), Jul 16 2004
p^2-1=12*sum_{i=0..floor(p/4)} floor[sqrt(i*p)] where p=a(n)=4n+1 [Shirali].
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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
.................................
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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];
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MATHEMATICA
| Select[4*Range[140] + 1, PrimeQ[ # ] &] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 16 2006
aa = {}; Do[If[Mod[Prime[n], 4] == 1, AppendTo[aa, Prime[n]]], {n, 1, 200}]; aa [From Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008]
lst={}; Do[Do[p=n^2+m^2; If[PrimeQ[p], AppendTo[lst, p]], {n, 0, 5!}], {m, 0, 5!}]; lst; Take[Union[lst], 123] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 21 2009]
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PROG
| (PARI) select(primes(1000), p->p%4==1) \\ version 2.4.2 or before
select(p->p%4==1, primes(1000)) \\ newer versions
(Haskell)
a002144 n = a002144_list !! (n-1)
a002144_list = filter ((== 1) . (`mod` 4)) a000040_list
-- Reinhard Zumkeller, Feb 22 2011
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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.
Sequence in context: A191218 A077426 A175768 * A192592 A111055 A145016
Adjacent sequences: A002141 A002142 A002143 * A002145 A002146 A002147
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 21 2000
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