A391748 - OEIS

%I #110 Feb 23 2026 13:37:08

%S 2,3,6,18,69,85,163,273,295,409,1087,1677,1798,3027,7395

%N Indices k such that y^2 + x*y - x^2 is prime, where (x,y) is the k-th fundamental solution to the Pell equation x^2 - 2*y^2 = 1.

%C This sequence represents the iteration numbers (k) of the fundamental solutions to the Pell equation x^2 - 2*y^2 = 1 such that the associated "Geometric Prime Polynomial" P(x,y) = y^2 + x*y - x^2 results in a prime number. The k-th solution is defined by x_k + y_k*sqrt(2) = (3 + 2*sqrt(2))^k.

%F Integers k such that y_k^2 + x_k*y_k - x_k^2 is prime, where x_k + y_k*sqrt(2) = (3 + 2*sqrt(2))^k.

%e For k=2, (x,y) = (17, 12). P(17,12) = 12^2 + 1712 - 17^2 = 59, which is prime.

%e For k=3, (x,y) = (99, 70). P(99,70) = 70^2 + 9970 - 99^2 = 2029, which is prime.

%o (Python)

%o from sympy import isprime

%o def a(n_limit):

%o     x, y, k = 3, 2, 1

%o     res = []

%o     while len(res) < n_limit:

%o         candidate = (y**2) + (x * y) - (x**2)

%o         if isprime(candidate):

%o             res.append(k)

%o         x, y = 3*x + 4*y, 2*x + 3*y

%o         k += 1

%o     return res

%Y Cf. A001541 (x values), A001542 (y values).

%K nonn,hard,more

%O 1,1

%A _Oğuz Serdar Uzunlar_, Feb 15 2026

%E a(14)-a(15) found by Elif Beyza Özer and İrem Yıldız and added by _Oğuz Serdar Uzunlar_, Feb 23 2026