A262027 The positive fundamental solutions x = x0(n) for the Pell equation x^2 - d*y^2 = +1 with odd y = y0(n). Then d coincides with d(n) = A007970(n).

2, 8, 3, 10, 4, 170, 24, 5, 26, 1520, 17, 6, 19, 3482, 48, 7, 50, 530, 8, 48842, 3480, 26, 80, 9, 82, 28, 197, 1574, 49, 10, 227528, 51, 962, 1126, 120, 11, 122, 4730624, 577, 10610, 244, 35, 77563250, 12, 1728148040, 37, 1324, 721, 64080026, 168, 13, 170, 2024, 199, 4190210

Offset: 1

Author

Wolfdieter Lang, Oct 04 2015

Keywords

• nonn

Comments

The corresponding values y = y0(n) are given by A262026(n).

This is a proper subset of A033313 corresponding to D values from d(n) = A007970(n).

For the proof that d(n) = A007970(n), the products of Conway's 2-happy couples, see the W. Lang link under A007970.

If d(n) = A007970(n) is odd (necessarily congruent to 3 modulus 4) then x0(n) is even, and if d(n) is even (necessarily congruent to 0 modulus 8) then x0 is odd.

Examples

			For the first [d(n), x0(n), y0(n)] see A262026.
		

Crossrefs

Cf. A007970, A033313, A262026.

Formula

a(n)^2 - d(n)*y0(n)^2 = +1 with y0(n) = A262026(n) and d(n) = A007970(n). (x0(n) = a(n), y0(n)) are the positive fundamental solutions of this Pell equation x^2 - d*y^2 = +1 with odd y = y0.