A262026 The positive odd fundamental solutions y = y0(n) for the Pell equation x^2 - d*y^2 = +1. It turns out that d = d(n) coincides with A007970(n).

1, 3, 1, 3, 1, 39, 5, 1, 5, 273, 3, 1, 3, 531, 7, 1, 7, 69, 1, 5967, 413, 3, 9, 1, 9, 3, 21, 165, 5, 1, 22419, 5, 93, 105, 11, 1, 11, 419775, 51, 927, 21, 3, 6578829, 1, 140634693, 3, 105, 57, 5019135, 13, 1, 13, 153, 15, 313191, 123, 650783, 7, 1, 1153080099, 7, 45, 19162705353, 3, 33, 5

Offset: 1

Author

Wolfdieter Lang, Oct 04 2015

Keywords

• nonn

Comments

The corresponding x = x0(n) values are given by A262027(n).

This is a proper subset of A033317 corresponding to its odd members.

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

For the positive even fundamental solutions y = y0(n) of x^2 - d*y^2 = 1, where d = d(n) coincides with A007969(n) see 2*A261250(n).

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

			The first triples [d(n), x0(n), y0(n)] are: [3,2,1], [7,8,3], [8,3,1], [11,10,3], [15,4,1], [19,170,39], [23,24,5], [24,5,1], [27,26,5], [31,1520,273], [32,17,3], [35,6,1], [40,19,3], [43,3482,531], [47,48,7], [48,7,1], [51,50,7], [59,530,69], [63,8,1], [67,48842,5967], [71,3480,413], [75,26,3], [79,80,9], [80,9,1], [83,82,9], [87,28,3], [88,197,21], [91,1574,165], [96,49,5], [99,10,1], [103,227528,22419], ...
		

Crossrefs

Cf. A007970, A033317, A262027, A262028.

Formula

x0(n)^2 - d(n)*a(n)^2 = +1 with x0(n) =

A262027(n) and d(n) = A007970(n). (x0(n), y0(n) = a(n)) are the positive fundamental solutions of this Pell equation x^2 - d*y^2 = +1 with odd y = y0.