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A002313
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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)
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60
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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; internal format)
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OFFSET
| 1,1
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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 (sven-h.simon(AT)t-online.de), Nov 30 2003
Except for 2, primes of the form x^2+4y^2. See A140633. - T. D. Noe (noe(AT)sspectra.com), May 19 2008
Primes p such that for all p>2, p XOR 2 = p + 2. - Brad Clardy, Oct 25 2011
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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.
D. Cox, "Primes of 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).
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..1000
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].
Dario Alpern, Online program that calculates sum of two squares representation
Eric Weisstein's World of Mathematics, Fermat's 4n+1 Theorem
G. Xiao, Two squares
Index entries for Gaussian integers and primes
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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:
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MATHEMATICA
| Select[ Prime@ Range@ 115, Mod[#, 4] != 3 &] (* RGWv *)
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PROG
| (PARI) select(primes(1000), p->p%4!=3) \\ Charles R Greathouse IV, Feb 11 2011
\\ Reverse the arguments above for newer versions of GP
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CROSSREFS
| Apart from initial term, same as A002144. For values of x and y see A002330, A002331.
Cf. A033203, A038873, A038874, A045331, A008784, A057129, A084163, A084165, A002144, A137351.
Sequence in context: A109515 A135933 A086807 * A177349 A160215 A068486
Adjacent sequences: A002310 A002311 A002312 * A002314 A002315 A002316
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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 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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