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 A002313 Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p. (Formerly M1430 N0564) 127
 2, 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 Or, primes p such that x^2 - p*y^2 represents -1. Primes which are not Gaussian primes (meaning not congruent to 3 mod 4). Every Fibonacci prime (with the exception of F(4) = 3) is in the sequence. If p = 2n+1 is the prime index of the Fibonacci prime, then F(2n+1) = F(n)^2 + F(n+1)^2 is the unique representation of the prime as sum of two squares. - Sven Simon, Nov 30 2003 Except for 2, primes of the form x^2 + 4y^2. See A140633. - T. D. Noe, May 19 2008 Primes p such that for all p > 2, p XOR 2 = p + 2. - Brad Clardy, Oct 25 2011 Greatest prime divisor of r^2 + 1 for some r. - Michel Lagneau, Sep 30 2012 Empirical result: a(n), as a set, compose the prime factors of the family of sequences produced by A005408(j)^2 + A005408(j+k)^2 = (2j+1)^2 + (2j+2k+1)^2, for j >= 0, and a given k >= 1 for each sequence, with the addition of the prime factors of k if not already in a(n). - Richard R. Forberg, Feb 09 2015 Primes such that when r is a primitive root then p-r is also a primitive root. - Emmanuel Vantieghem, Aug 13 2015 Primes of the form (x^2 + y^2)/2. Note that (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2 = a^2 + b^2 with x = a + b and y = a - b. More generally, primes of the form (x^2 + y^2) / A001481(n) for every fixed n > 1. - Thomas Ordowski, Jul 03 2016 Numbers n such that ((n-2)!!)^2 == -1 (mod n). - Thomas Ordowski, Jul 25 2016 Primes p such that (p-1)!! == (p-2)!! (mod p). - Thomas Ordowski, Jul 28 2016 The product of 2 different terms (x^2 + y^2)(z^2 + v^2) = (xz + yv)^2 + (xv - yz)^2 is sum of 2 squares (A000404) because (xv - yz)^2 > 0. If x were equal to yz/v then (x^2 + y^2)/(z^2 + v^2) would be equal to ((yz/v)^2 + y^2)/(z^2 + v^2) = y^2/v^2 which is not possible because (x^2 + y^2) and (z^2 + v^2) are prime numbers. For example, (2^2 + 5^2)(4^2 + 9^2) = (2*4 + 5*9)^2 + (2*9 - 5*4)^2. - Jerzy R Borysowicz, Mar 21 2017 REFERENCES M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 872. David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 251, 252. 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 Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe) Dario Alpern, Online program that calculates sum of squares representation N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528. Eric Weisstein's World of Mathematics, Fermat's 4n+1 Theorem G. Xiao, Two squares Index entries for Gaussian integers and primes FORMULA a(n) ~ 2n log n. - Charles R Greathouse IV, Jul 04 2016 a(n) = A002331(n)^2 + A002330(n)^2. See crossrefs. - Wolfdieter Lang, Dec 11 2016 EXAMPLE 13 is in the sequence since it is prime and 13 = 4*3 + 1. Also 13 = 2^2 + 3^2. And -1 is a square (mod 13): -1 + 2*13 = 25 = 5^2. Of course, only the first term is congruent to 2 (mod 4). - Michael B. Porter, Jul 04 2016 MAPLE with(numtheory): for n from 1 to 300 do if ithprime(n) mod 4 = 1 or ithprime(n) mod 4 = 2 then printf(`%d, `, ithprime(n)) fi; od: # alternative A002313 := proc(n) option remember ; local a; if n = 1 then 2; elif n = 2 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: seq(A002313(n), n=1..100) ; # R. J. Mathar, Feb 01 2024 MATHEMATICA Select[ Prime@ Range@ 115, Mod[#, 4] != 3 &] (* Robert G. Wilson v *) fQ[n_] := Solve[x^2 + 1 == n*y^2, {x, y}, Integers] == {}; Select[ Prime@ Range@ 115, fQ] (* Robert G. Wilson v, Dec 19 2013 *) PROG (PARI) select(p->p%4!=3, primes(1000)) \\ Charles R Greathouse IV, Feb 11 2011 (Haskell) a002313 n = a002313_list !! (n-1) a002313_list = filter ((`elem` [1, 2]) . (`mod` 4)) a000040_list -- Reinhard Zumkeller, Feb 04 2014 (Magma) [p: p in PrimesUpTo(700) | p mod 4 in {1, 2}]; // Vincenzo Librandi, Feb 18 2015 CROSSREFS Apart from initial term, same as A002144. For values of x and y see A002330 and A002331. Cf. A033203, A038873, A038874, A045331, A008784, A057129, A084163, A084165, A137351. Sequence in context: A109515 A135933 A086807 * A233346 A182198 A291275 Adjacent sequences: A002310 A002311 A002312 * A002314 A002315 A002316 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane EXTENSIONS More terms from Henry Bottomley, Aug 10 2000 More terms from James A. Sellers, Aug 22 2000 STATUS approved

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