

A325082


Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + x*y + 69*y^2.


3



89, 179, 419, 449, 599, 619, 709, 719, 829, 859, 1039, 1109, 1259, 1489, 1549, 1709, 1879, 2039, 2099, 2179, 2539, 2579, 2689, 2909, 3169, 3259, 3359, 3389, 3499, 3919, 4019, 4159, 4229, 4349, 4409, 4799, 4909, 5009, 5039, 5179, 5449, 5569, 5659, 5779, 5839
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Brink showed that prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 are representable by exactly one of the quadratic forms x^2 + x*y + 14*y^2 or x^2 + x*y + 69*y^2. A325081 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.


LINKS

Table of n, a(n) for n=1..45.
David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129(2) (2009), 464468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.
Rémy Sigrist, PARI program for A325082
Wikipedia, Kaplansky's theorem on quadratic forms


EXAMPLE

Regarding 2099:
 2099 is a prime number,
 2099 = 38*55 + 9,
 2099 = 17^2 + 1*17*5 + 69*5^2,
 hence 2099 belongs to this sequence.


PROG

(PARI) See Links section.


CROSSREFS

See A325067 for similar results.
Cf. A325081.
Sequence in context: A106758 A142335 A230168 * A033670 A044421 A044802
Adjacent sequences: A325079 A325080 A325081 * A325083 A325084 A325085


KEYWORD

nonn


AUTHOR

Rémy Sigrist, Mar 28 2019


STATUS

approved



