%I A002144 M3823 N1566
%S A002144 5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,149,157,173,181,193,
%T A002144 197,229,233,241,257,269,277,281,293,313,317,337,349,353,373,389,397,
%U A002144 401,409,421,433,449,457,461,509,521,541,557,569,577,593,601,613,617
%N A002144 Pythagorean primes: primes of form 4n+1.
%C A002144 These are the primitive elements of A009003.
%C A002144 -1 is a quadratic residue mod a prime p iff p is in this sequence.
%C A002144 sin(a(n)*pi/2) = 1 with pi=3.1415..., see A070750. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), May 04 2002
%C A002144 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
%C A002144 Odd primes such that binomial(p-1,(p-1)/2) == 1 (mod p) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Feb 07 2004
%C A002144 Primes that are the hypotenuse of a right triangle with integer sides. 
               The Pythagorean triple is {A002365(n+4), A002366(n+4),a(n)}.
%C A002144 Also, primes of the form a^k + b^k, k >1 (cf. A089716). - Amarnath Murthy 
               (amarnath_murthy(AT)yahoo.com), Nov 17 2003
%C A002144 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
%C A002144 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
%C A002144 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
%C A002144 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]
%C A002144 A079260(a(n)) = 1; complement of A137409. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 11 2008]
%C A002144 Contribution from Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008: (Start)
%C A002144 If we take 4 numbers : 1, A002314(n), A152676(n), A152680(n) then
%C A002144 multiplication table modulo A002144(n) is isomorphc to the Latin square:
%C A002144 1 2 3 4
%C A002144 2 4 1 3
%C A002144 3 1 4 2
%C A002144 4 3 2 1
%C A002144 and isomorphic to the multiplication table of {1, I, -I, -1} where I 
               is Sqrt[ -1],
%C A002144 A152680(n) is isomorphic to -1, A002314(n) with I or -I and A152676(n) 
               vice versa -I or I.
%C A002144 1, A002314(n), A152676(n), A152680(n) are subfield of Galois Field [A002144(n)]. 
               (End)
%C A002144 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]
%D A002144 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002144 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002144 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 870.
%D A002144 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%D A002144 M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 
               2003; see p. 76.
%D A002144 S. A. Shirali, A family portrait of primes-a case study in discrimination, 
               Math. Mag. 70 (4) (1997) 263.
%H A002144 T. D. Noe, <a href="b002144.txt">Table of n, a(n) for n = 1..1000</a>
%H A002144 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A002144 C. Banderier, <a href="http://algo.inria.fr/banderier/Recipro/node14.html">
               Calcul de (-1/p)</a>
%H A002144 J. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/
               miniature8.pdf">The Quadratic Residue Theorem</a>
%H A002144 R. Chapman, <a href="http://www.maths.ex.ac.uk/~rjc/courses/nt03/quadrec.pdf">
               Quadratic reciprocity</a>
%H A002144 J. E. Ewell, <a href="http://www.math.buffalo.edu/mad/work/paper.ewell.johna.html">
               A Simple Proof of Fermat's Two-Square Theorem</a>
%H A002144 A. Granville and G. Martin, <a href="http://www.arXiv.org/abs/math.NT/
               0408319">Prime number races</a>
%H A002144 D. & C. Hazzlewood, <a href="http://www.math.swt.edu/~haz/prob_sets/notes/
               node32.html">Quadratic Reciprocity</a>
%H A002144 R. C. Laubenbacher & D. J. Pengelley, <a href="http://math.nmsu.edu/~history/
               eisenstein/eisenstein.html">Eisenstein's Misunderstood Geometric 
               Proof of the Quaratic Reciprocity Theorem</a>
%H A002144 R. C. Laubenbacher & D. J. Pengelley, <a href="http://math.nmsu.edu/~history/
               schauspiel/schauspiel.html">Gauss, Eisenstein and the -third' proof 
               of the Quadratic Reciprocity Theorem</a>
%H A002144 K. Matthews, <a href="http://www.numbertheory.org/php/serret.html">Serret's 
               algorithm based Server</a>
%H A002144 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WilsonsTheorem.html">Wilson's Theorem</a>
%H A002144 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PythagoreanTriple.html">Pythagorean Triples</a>
%H A002144 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/
               JacobiSymbol/31/01/ShowAll.html">The Gauss Reciprocity Law</a>
%H A002144 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">
               Two squares</a>
%H A002144 Wikipedia, <a href="http://en.wikipedia.org/wiki/Quadratic_reciprocity">
               Quadratic reciprocity</a>
%F A002144 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
%F A002144 p^2-1=12*sum_{i=0..floor(p/4)} floor[sqrt(i*p)] where p=a(n)=4n+1 [Shirali].
%e A002144 The following table shows the relationship
%e A002144 between several closely related sequences:
%e A002144 Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
%e A002144 a = A002331, b = A002330, t_1 = ab/2 = A070151;
%e A002144 p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
%e A002144 t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
%e A002144 with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
%e A002144 ---------------------------------
%e A002144 .p..a..b..t_1..c...d.t_2.t_3..t_4
%e A002144 ---------------------------------
%e A002144 .5..1..2...1...3...4...4...3....6
%e A002144 13..2..3...3...5..12..12...5...30
%e A002144 17..1..4...2...8..15...8..15...60
%e A002144 29..2..5...5..20..21..20..21..210
%e A002144 37..1..6...3..12..35..12..35..210
%e A002144 41..4..5..10...9..40..40...9..180
%e A002144 53..2..7...7..28..45..28..45..630
%e A002144 .................................
%p A002144 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];
%t A002144 Select[4*Range[140] + 1, PrimeQ[ # ] &] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 16 2006
%t A002144 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]
%t A002144 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]
%Y A002144 For values of n see A005098. Cf. A002145, A002476. Apart from initial 
               term, same as A002313.
%Y A002144 Cf. A114200.
%Y A002144 Cf. A003658.
%Y A002144 A002314, A152676, A152680. [From Artur Jasinski (grafix(AT)csl.pl), Dec 
               10 2008]
%Y A002144 Sequence in context: A078900 A113482 A077426 this_sequence A111055 A145016 
               A123079
%Y A002144 Adjacent sequences: A002141 A002142 A002143 this_sequence A002145 A002146 
               A002147
%K A002144 nonn,easy,nice
%O A002144 1,1
%A A002144 N. J. A. Sloane (njas(AT)research.att.com).
%E A002144 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 21 2000