The Pellian Equation x² −Dy² = 1 from a Sequence S_n (Part XVII)

A Method for Generating Pellian Triples (x,y,1)

The Pellian equation x² − Dy² = 1 was covered in Part I and the negative Pellian equation x² − Dy² = −1 in Part II. The least solutions of the negative Pell equation, however, are not posted in either Wikipedia (which has a small section describing this topic) or listed in Recreations in the Theory of Numbers by Albert H. Beiler (1966) as were their positive Pell solutions, but the following equations on page 253 may be used for their computation:

In addition, the method of converting a quadratic surd √D into continued fractions (pages 261-262) are also methods that can be used to generate these least solutions. On the other hand, this method is now available on this website as a computer program making it extremely easy to generate the convergents. As to the topic of this article, it concerns the sequence S_n generated from the the equation 4(n)(n+3) + 5 which was found in the OEIS database with the Sloane number A078371 as the equation (2n + 5)(2n + 1). Note the equations though dissimilar are out of phase by one, i.e., n = 1 in the former is the same as n = 0 in the latter. The comment section describes the numbers in the sequence as quote "generic form of D in the (nontrivially) solvable Pell equation x² + y²D = +4":

These D values entered into the Pell calculator program gives initial values of Q_n = +4 at n = 3 and 5 which leads to x and y values at positions n = 2 and 4. The Sloane comment that Q_n is +4 is confusing and most important it makes no mention that the Q_n = +1 for the numbers in this series all fall initially at n = 7 and multiples of 6m + 7 with m ≥ 0. In addition, all x and y are situated at n = 6 (and also at multiples of 6) as the variables p_n and q_n. See the examples for 117 at Code 117 and 1221 at Code 1221. The red number 5 in the above sequence although generated by the equation does not share the properties of the numbers under discussion since its Q_n is 1 for all n (see Code 5).

Table I shows the first 13 D numbers in the sequence S_n.

The Pell equations are set up according to the following formats with the x and ys taken from Table I. The first equation with n₁ and n₂ gives the alternate expression for x + y√D where the square term is divided by ½n₁ and n₂ is a triangular number n_t equal to (n² + n)/2.

Table II gives the formulas for all 13 Pell equations according to the format listed above where both formulas on the right are equivalent. Entry one with D = 5 is added here to show that n₂ = 1 is the first triangular number although the equation is not part of the group as stated previously. The values for p_n and q_n were obtained from the first least solution at n = 2. The triangular numbers are in blue.

This concludes Part XVII. To go to Part XVIII where the triangular numbers are used in the calculation of n₁.

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