The Pellian Equation x² −Dy² = ±1 from the Sequence (n + 1)² + 1 (Part XIV)

The Pellian equation is the Diophantine equation x² − Dy² = z² where z equals 1. The least solutions of the Pell equation are posted in Wikipedia. and also listed in Table 91, page 254 of Recreations in the Theory of Numbers by Albert H. Beiler (1966), where the values for D on page 252-253 have been computed using the following two expressions:

Bailer also describes that when Q_n is 1 the Pell equation x² −Dy² = −1 is soluble. In fact this can be verified using series of equations on this website which can be used to calculate for the x and y values of a Pell equation. It is known that the sequence S_m having the OEIS number A002522 containing the equation (n + 1)² + 1 is the continued fraction expansion of sqrt(n) and thus can produce a subset of the negative Pell values from which the following first fourteen values have been extracted:

In addition, to see which values correpond to the negative Pell equation see A031396 on the OEIS website.

The numbers corresponding to the above sequence can be used to construct tables similar to those constructed in Part IV. Instead I'll show least solution values for Code 10 and Code 101 using the appropriate computer programs so that one can visualize the similarity between these two results and get a taste for the least solution calculations. All numbers within this subgroup behave similarly and the secondary, tertiary, etc., values for x and y as p_n and q_n, respectively, can be seen to increase to larger and larger values.

This concludes Part XIV. Go to Part IV.

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