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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)
71
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

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

T. D. Noe and Zak Seidov, Table of n, a(n) for n = 1..10000  (first 1000 terms from Noe).

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

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

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:

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(primes(1000), p->p%4!=3) \\ Charles R Greathouse IV, Feb 11 2011

\\ Reverse the arguments above for newer versions of GP

(Haskell)

a002313 n = a002313_list !! (n-1)

a002313_list = filter ((`elem` [1, 2]) . (`mod` 4)) a000040_list

-- Reinhard Zumkeller, Feb 04 2014

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 * A233346 A182198 A177349

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 and James A. Sellers, Aug 22 2000

STATUS

approved

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Last modified August 23 02:55 EDT 2014. Contains 245976 sequences.