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A005471
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Primes of the form m^2 + 3m + 9, where m can be positive or negative.
(Formerly M4345)
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29
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7, 13, 19, 37, 79, 97, 139, 163, 313, 349, 607, 709, 877, 937, 1063, 1129, 1489, 1567, 1987, 2557, 2659, 3313, 3547, 4297, 5119, 5557, 7489, 8017, 8563, 9127, 9319, 9907, 10513, 11779, 12889, 15013, 15259, 16519, 17299, 18097, 18367, 18913, 20029
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OFFSET
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1,1
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COMMENTS
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Primes of the form m^2 + m + 7, for some m >= 0. - Daniel Forgues, Jan 26 2020
Primes p such that 4*p - 27 is a square. Also, primes p such that the Galois group of the polynomial X^3 - p*X + p over Q is the cyclic group of order 3. See Conrad, Corollary 2.5. - Peter Bala, Oct 17 2021
Primes p such that the Galois group of the cubic X^3 + p*(X + 1)^2 over Q is the cyclic group C_3.
If p = m^2 + 3*m + 9 is prime then the Galois group of the cubic X^3 - m*X^2 - (m + 3)*X - 1 over Q is C_3. See Shanks.
The pair of cubics X^3 - m*p*X^2 - 3*(m+1)*p*X - (2*m+3)*p and X^3 - 2*p*X^2 + p*(p - 10)*X + p*(p - 8) also have their Galois groups over Q equal to C_3 (both cubics are irreducible over Q by Eisenstein's criteria). Apply Conrad, Corollary 2.5. (End)
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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).
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LINKS
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FORMULA
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EXAMPLE
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For m = -11, -10, ..., 22 the primes of the form m^2+3m+9 are 97, 79, 37, 19, 13, 7, 7, 13, 19, 37, 79, 97, 139, 163, 313, 349.
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MAPLE
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if n = 1 then
7;
else
end if;
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MATHEMATICA
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Select[Table[n^2 + 3*n + 9, {n, -1, 200}], PrimeQ] (* T. D. Noe, Mar 21 2013 *)
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PROG
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(Magma) [a: n in [-1..150] | IsPrime(a) where a is n^2+3*n+9]; // Vincenzo Librandi, Mar 22 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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