|
| |
|
|
A274675
|
|
Primes p such that p = x^2 + 14*y^2 or p = 2*x^2 + 7*y^2, where p != 2, 7 and x, y are integers.
|
|
1
|
|
|
|
23, 71, 79, 113, 127, 137, 151, 191, 193, 233, 239, 263, 281, 337, 359, 401, 431, 449, 457, 463, 487, 569, 599, 617, 631, 641, 673, 743, 751, 809, 823, 863, 911, 919, 953, 967, 977, 991, 1009, 1031, 1033, 1087, 1103, 1129, 1201, 1289, 1297, 1303, 1327
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Also primes congruent to {1, 9, 15, 23, 25, 39} mod 56.
Discriminant -8*7. The product of two entries is congrunet to {1, 7} (mod 8). Buell, p. 51, 3).
The given two reduced positive definite binary quadratic forms represent the odd primes, not 7, with the generic characters Legendre(p|7) = +1 and Legendre(2|p) = +1. The other two reduced forms are [3, 2, 5] and [3,-2, 5] with values -1 and -1 for these two generic characters, and give the odd primes, not 7, listed in A106915. This is related to the two genera of discriminant -56 with class number h(-56) = 4. See Buell, p. 52, 2), and Cox, p. 30.
There is a misprint 29 (instead of 39) in Cox (1989, ISBN 0-471-50654-0), p. 33, in eqs. (2.21) and 2.23). (End)
In the first US edition, there just one error, in Eq. (2.21), and it is on page 33. In the second edition this error has been corrected. - N. J. A. Sloane, Jun 04 2021
|
|
|
REFERENCES
|
D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 51-53.
David A. Cox, Primes of the Form x^2 + n y^2, John Wiley & Sons, 1st edition, 1989; 2nd edition, 2003.
|
|
|
LINKS
|
|
|
|
MATHEMATICA
|
Select[Prime@Range[300], MemberQ[{1, 9, 15, 23, 25, 39}, Mod[#, 56]] &]
|
|
|
PROG
|
(Magma) [p: p in PrimesUpTo(3000) | p mod 56 in {1, 9, 15, 23, 25, 39} ];
(PARI) is(n) = ispseudoprime(n) && #setintersect([n % 56], [1, 9, 15, 23, 25, 39])==1 \\ Felix Fröhlich, Jul 02 2016
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
