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.

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

Offset: 1

Author

Vincenzo Librandi, Jul 02 2016

Keywords

• nonn
• easy

Comments

Also primes congruent to {1, 9, 15, 23, 25, 39} mod 56.

From Wolfdieter Lang, Jun 04 2021: (Start)

Discriminant -8*7. The product of two entries is congruent 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

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Thomas R. Hagedorn, Primes of the form x^2+ny^2 and the geometry of (convenient) numbers (page 3).

Crossrefs

Cf. A033202, A106915.

Programs

Magma

[p: p in PrimesUpTo(3000) | p mod 56 in {1, 9, 15, 23, 25, 39} ];

Mathematica

Select[Prime@Range[300], MemberQ[{1, 9, 15, 23, 25, 39}, Mod[#, 56]] &]

PARI

is(n) = ispseudoprime(n) && #setintersect([n % 56], [1, 9, 15, 23, 25, 39])==1 \\ Felix Fröhlich, Jul 02 2016