Norm Forms of Pell Numbers: The Sequence P_k^2 - 2

We study the sequence D_k = P_k^2 - 2, arising as the norm N(P_k + sqrt(2)) in Z[sqrt(2)]. We prove: (1) D_k is prime only if k is odd (mod 8 obstruction); (2) an odd prime p divides some D_k only if p ≡ ±1 (mod 8), and this is not sufficient; (3) the continued fraction of sqrt(n^2-2) is [n-1; 1, n-2, 1, 2n-2] with period 4, yielding that the fundamental solution of x^2-(n^2-2)y^2=1 is (n^2-1, n). Class numbers h(Q(sqrt(D_k))) are computed for prime values: h=1, 3, 15 for k=3, 5, 7. A Mersenne-type conjecture on infinitely many prime values is stated.