

A005471


Primes of the form m^2 + 3m + 9, where m can be positive or negative.
(Formerly M4345)


29



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

1,1


COMMENTS

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)


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA



EXAMPLE

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.


MAPLE

if n = 1 then
7;
else
end if;


MATHEMATICA

Select[Table[n^2 + 3*n + 9, {n, 1, 200}], PrimeQ] (* T. D. Noe, Mar 21 2013 *)


PROG

(Magma) [a: n in [1..150]  IsPrime(a) where a is n^2+3*n+9]; // Vincenzo Librandi, Mar 22 2013


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



