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 A002144 Pythagorean primes: primes of form 4*k + 1. (Formerly M3823 N1566) 461
 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 Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017 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. - 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 E. Stange, Feb 03 2010 Subsequence of A007969. - Reinhard Zumkeller, Jun 18 2011 A151763(a(n)) = 1. 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 statement that these primes are the sum of two nonzero squares follows from Fermat's theorem on the sum of two squares. - Jerzy R Borysowicz, Jan 02 2019 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 This is 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 This is the subsequence of primes of A004431 and also of A016813. - Bernard Schott, Apr 30 2022 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 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). LINKS T. D. Noe and Zak Seidov, 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. Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102. C. Banderier, Calcul de (-1/p) J. Butcher, Mathematical Miniature 8: The Quadratic Residue Theorem, NZMS Newsletter, No. 75, April 1999. Hing Lun Chan, Windmills of the minds: an algorithm for Fermat's Two Squares Theorem, arXiv:2112.02556 [cs.LO], 2021. R. Chapman, Quadratic reciprocity A. David Christopher, A partition-theoretic proof of Fermat’s Two Squares Theorem, Discrete Mathematics, Volume 339, Issue 4, 6 April 2016, Pages 1410-1411. 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 Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33. Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018. R. C. Laubenbacher and D. J. Pengelley, Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem R. C. Laubenbacher and D. J. Pengelley, Gauss, Eisenstein and the 'third' proof of the Quadratic Reciprocity Theorem K. Matthews, Serret's algorithm based Server Carlos Rivera, Puzzle 968. Another property of primes 4m+1, The Prime Puzzles & Problems Connection. 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 Wikipedia, Quadratic reciprocity Wolfram Research, The Gauss Reciprocity Law G. Xiao, Two squares 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)] 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) = 4*n + 1. [Shirali] a(n) = A000290(A002972(n)) + A000290(2*A002973(n)) = A000290(A002331(n+1)) + A000290(A002330(n+1)). - Reinhard Zumkeller, Feb 16 2010 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 a(n) = 4*A005098(n) + 1. - Zak Seidov, Sep 16 2018 From Vaclav Kotesovec, Apr 30 2020: (Start) Product_{k>=1} (1 - 1/a(k)^2) = A088539. Product_{k>=1} (1 + 1/a(k)^2) = A243380. Product_{k>=1} (1 - 1/a(k)^3) = A334425. Product_{k>=1} (1 + 1/a(k)^3) = A334424. Product_{k>=1} (1 - 1/a(k)^4) = A334446. Product_{k>=1} (1 + 1/a(k)^4) = A334445. Product_{k>=1} (1 - 1/a(k)^5) = A334450. Product_{k>=1} (1 + 1/a(k)^5) = A334449. (End) From Vaclav Kotesovec, May 05 2020: (Start) Product_{k>=1} (1 + 1/A002145(k)) / (1 + 1/a(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962... Product_{k>=1} (1 - 1/A002145(k)) / (1 - 1/a(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End) Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log((2*n*s)! * zeta(n*s) * abs(EulerE(n*s - 1)) / (Pi^(n*s) * 2^(2*n*s) * BernoulliB(2*n*s) * (2^(n*s) + 1) * (n*s - 1)!))/n, s >= 3 odd number. - Dimitris Valianatos, May 21 2020 Legendre symbol (-1, a(n)) = +1, for n >= 1. - Wolfdieter Lang, Mar 03 2021 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 ................................. 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 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 + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 16 2006 *) Select[Prime[Range], Mod[#, 4]==1&] (* Harvey P. Dale, Jan 28 2021 *) PROG (Haskell) a002144 n = a002144_list !! (n-1) a002144_list = filter ((== 1) . a010051) [1, 5..] -- Reinhard Zumkeller, Mar 06 2012, Feb 22 2011 (Magma) [a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014 (PARI) select(p->p%4==1, primes(1000)) (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 (Python) from sympy import isprime print(list(filter(isprime, range(1, 618, 4)))) # Michael S. Branicky, May 13 2021 (Sage) def A002144_list(n): # returns all Pythagorean primes <= n     return [x for x in prime_range(5, n+1) if x % 4 == 1] A002144_list(617) # Peter Luschny, Sep 12 2012 CROSSREFS Cf. A002145, A002314, A002476, A002972, A002973, A003658, A004431, A007519, A010051, A016813, A076339, A094407. Cf. A114200, A133870, A142925, A152676, A152680, A173330, A173331, A208177, A208178, A334912. Cf. A004613 (multiplicative closure). Apart from initial term, same as A002313. For values of n see A005098. Primes in A020668. Sequence in context: A231754 A175768 A351535 * A280084 A319287 A192592 Adjacent sequences:  A002141 A002142 A002143 * A002145 A002146 A002147 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified October 5 14:51 EDT 2022. Contains 357259 sequences. (Running on oeis4.)