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Search: id:A002313
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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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+0
49
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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; listen)
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OFFSET
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1,1
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COMMENT
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
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LINKS
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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 Plus 1 Theorem
G. Xiao, Two squares
Index entries for Gaussian integers and primes
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MAPLE
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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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CROSSREFS
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Apart from initial term, same as A002144. For values of x and y see A002330, A002331.
Cf. A033203, A038873, A038874, A045331, A008784, A057129.
Cf. A084163, A084165, A002144, A137351.
Adjacent sequences: A002310 A002311 A002312 this_sequence A002314 A002315 A002316
Sequence in context: A109515 A135933 A086807 this_sequence A160215 A068486 A099332
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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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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