Properties of the Pellian Sequence

The purpose of this thesis is to show that the Pellian sequence possesses a great deal of symmetry and regularity. The English method shall be introduced to show that there is a least positive integer x that makes Ax^+i a square if and only if A is a non-square. This method will be shown to produce the entire infinite solution set for any positive integer A.The method of continued fractions will be described and will be used to construct a simple algorithm to produce the Pellian sequence given in both the table and the diskette of Pellian numbers. It will also be shown that an intimate relationship holds between this method and the English method.Finally, properties of the Pellian sequence will be proved to demonstrate regularity in the sequence. It will be shown that every non-negative integer occurs infinitely often in the sequence and, for some classes of integers, the complete set of occurrences in the sequence will be determined prior to solving the Pell equation. A strong connection will be shown to hold between a given number's continued fraction expansion and its set of occurrences. Lastly, it will be conjectured that a given number is a prime power according to its first occurrence in the sequence.