OFFSET
1,1
COMMENTS
Primes of the form m^2 + 2*k^2 are the norms of prime elements of Z[i*sqrt(2)]. Pairs of primes of the form (m^2 + 2*k^2, m^2 + 2*(k+1)^2) correspond to primes in Z[i*sqrt(2)] differing by i*sqrt(2).
A prime cannot simultaneously be the lesser of such a pair and the greater of another.
FORMULA
If m^2 + 2*k^2 and m^2 + 2*(k+1)^2 are primes, then m == 3 (mod 6) and k == 1 (mod 3).
EXAMPLE
The first 3 such prime pairs are
(11,17) = (3^2 + 2*1^2, 3^2 + 2*2^2) with m=3 and k=1,
(41,59) = (3^2 + 2*4^2, 3^2 + 2*5^2) with m=3 and k=4,
(83,89) = (9^2 + 2*1^2, 9^2 + 2*2^2) with m=9 and k=1.
CROSSREFS
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Jan 28 2023
STATUS
approved