Pell's Equation/Sequence

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Sequence of Minimum Solutions to Pell's Equation

The minimum solutions of $x$ and $y$ for the Diophantine equation $x^2 - n y^2 = 1$ for various values of $n$ are as follows:

$n$ $x$ $y$
$2$ $3$ $2$
$3$ $2$ $1$
$5$ $9$ $4$
$6$ $5$ $2$
$7$ $8$ $3$
$8$ $3$ $1$
$10$ $19$ $6$
$11$ $10$ $3$
$12$ $7$ $2$
$13$ $649$ $180$
$14$ $15$ $4$
$15$ $4$ $1$

Note the unusual peak at $13$.


Thus, the sequence for $x$ for non-square $n$ begins:

$3, 2, 9, 5, 8, 3, 19, 10, 7, 649, 15, 4, 33, 17, 170, \ldots$

This sequence is A033313 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Similarly, the sequence for $y$ for non-square $n$ begins:

$2, 1, 4, 2, 3, 1, 6, 3, 2, 180, 4, 1, 8, 4, 39, 2, 12, \ldots$

This sequence is A033317 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Sources