Triangular Numbers and the Pellian Equation x² −Dy² = 1 (Part XX)

A New Sequence S_n for the Pellian

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 36(n)(n+1) − 9 (not in the OEIS database):

This sequence is of the type N(n)(n+1) + m employed in Part XIX but differs in that the D values entered into the Pell calculator program produce Q_n = +1 at three different n's instead of the same n as was shown in Part XIX. These ns may be considered as a triple (9, 5, 13) where the least solution all fall at the first n − 1 of Q_n. Although 9 and 13 give the desired solutions, it is the second least solution of 5, the one where n = 9 that is of interest, since the numbers at n − 1 of Q_n match up with those of n = 9 and n = 13. Code 711 shows the calculated convergents for D = 711 with its Q_n values of 1 at 5, 9, 13, 17, etc. This may be compared to the other n terms in the triple 9 and 13 Code 423 and Code 1071 where again the values of x and y fall to the left of Q_n at n − 1.

Table I shows the first 2-15 D terms in the sequence S_n not including the −9. As mentioned above every three n is a triple (9,[5,9],13) modified to show that it is the 2nd term in the bracket that is to be used. Moreover, the first with a Q_n = 3 and the second, the odd man out, are included since both fit the Pellian equation but don't fit the triplet rule.

The Pell equations are set up according to the following formats with the x and ys taken from Table I. The first equation with 8(n_t)² and n₂ gives the alternate expression for x + y√D where n_t is a triangular number equal to (n² + n)/2 and n₂ = y/16

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 = 207 is added here to show that n₂ = 3 is the second 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 = 8, n = 8 and n = 12, respectively, for each triple. The triangular numbers are in blue.

This concludes Part XX.

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