A391748 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.

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..15.

Formula

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.

Example

For k=2, (x,y) = (17, 12). P(17,12) = 12^2 + 1712 - 17^2 = 59, which is prime.
For k=3, (x,y) = (99, 70). P(99,70) = 70^2 + 9970 - 99^2 = 2029, which is prime.

Prog

(Python)
from sympy import isprime
def a(n_limit):
    x, y, k = 3, 2, 1
    res = []
    while len(res) < n_limit:
        candidate = (y**2) + (x * y) - (x**2)
        if isprime(candidate):
            res.append(k)
        x, y = 3*x + 4*y, 2*x + 3*y
        k += 1
    return res

Crossrefs

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

Sequence in context: A089424 A083563 A038056 * A072241 A093468 A185625

Adjacent sequences: A391745 A391746 A391747 * A391749 A391750 A391751

Keyword

nonn,hard,more

Author

Oğuz Serdar Uzunlar, Feb 15 2026

Extensions

a(14)-a(15) found by Elif Beyza Özer and İrem Yıldız and added by Oğuz Serdar Uzunlar, Feb 23 2026

Status

approved