OFFSET

1,1

COMMENTS

Kaplansky showed that prime numbers congruent to 1 modulo 16 are representable by both or neither of the quadratic forms x^2 + 32*y^2 and x^2 + 64*y^2. This sequence corresponds to those representable by both, and A325068 corresponds to those representable by neither.

Also, Kaplansky showed that prime numbers congruent to 9 modulo 16 are representable by exactly one of these quadratic forms. A325069 corresponds to those representable by the first form and A325070 to those representable by the second form.

Brink provided similar results for other congruences.

LINKS

David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129(2) (2009), 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.

Rémy Sigrist, PARI program for A325067

Wikipedia, Kaplansky's theorem on quadratic forms

EXAMPLE

Regarding 1201:

- 1201 is a prime number,

- 1201 = 75*16 + 1,

- 1201 = 7^2 + 32*6^2 = 25^2 + 64*3^2,

- hence 1201 belongs to the sequence.

PROG

(PARI) See Links section.

CROSSREFS

KEYWORD

nonn

AUTHOR

Rémy Sigrist, Mar 27 2019

STATUS

approved