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
- Weisstein, Eric W. "Pell Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PellEquation.html