Search: id:A002313
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%I A002313 M1430 N0564
%S A002313 2,5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,149,157,173,181,193,
%T A002313 197,229,233,241,257,269,277,281,293,313,317,337,349,353,373,389,397,
%U A002313 401,409,421,433,449,457,461,509,521,541,557,569,577,593,601,613,617
%N A002313 Primes congruent to 1 or 2 modulo 4; or, primes of form x^2+y^2; or, 
               -1 is a square mod p.
%C A002313 Or, primes p such that x^2 - p*y^2 represents -1.
%C A002313 Primes which are not Gaussian primes (meaning not congruent to 3 mod 
               4).
%C A002313 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 (sven-h.simon(AT)t-online.de), Nov 
               30 2003
%C A002313 Except for 2, primes of the form x^2+4y^2. See A140633. - T. D. Noe (noe(AT)sspectra.com), 
               May 19 2008
%D A002313 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002313 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002313 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.
%D A002313 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%D A002313 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.
%D A002313 J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 
               (1949), 517-528.
%H A002313 T. D. Noe, <a href="b002313.txt">Table of n, a(n) for n = 1..1000</a>
%H A002313 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 A002313 Dario Alpern, <a href="http://www.alpertron.com.ar/ENGLISH.HTM">Online 
               program that calculates sum of two squares representation</a>
%H A002313 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Fermats4nPlus1Theorem.html">Fermat's 4n Plus 1 Theorem</a>
%H A002313 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">
               Two squares</a>
%H A002313 <a href="Sindx_Ga.html#gaussians">Index entries for Gaussian integers 
               and primes</a>
%p A002313 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:
%Y A002313 Apart from initial term, same as A002144. For values of x and y see A002330, 
               A002331.
%Y A002313 Cf. A033203, A038873, A038874, A045331, A008784, A057129.
%Y A002313 Cf. A084163, A084165, A002144, A137351.
%Y A002313 Sequence in context: A109515 A135933 A086807 this_sequence A160215 A068486 
               A099332
%Y A002313 Adjacent sequences: A002310 A002311 A002312 this_sequence A002314 A002315 
               A002316
%K A002313 nonn,easy,nice
%O A002313 1,1
%A A002313 N. J. A. Sloane (njas(AT)research.att.com).
%E A002313 More terms from Henry Bottomley (se16(AT)btinternet.com), Aug 10 2000 
               and James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000

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