Triangular and the Odd Numbers and the Pellian Equation(Part XIXC)

Continuation:A Method for Generating Pellian Triples

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.

This article, a continuation of Part XIXB, is in regard to the sequences S_n generated from the the equations of the type N(n)(n+1) + m which have very interesting properties as was shown in Part XVII for 4(n)(n+3) + 5 and in Part IIIA for 4(n)(n+1) + 5. Six other sequences with similar equations are displayed in Part XIXB and these two new sequences in this article are part of this group, in numerical order:

where all the sequences were evaluated at n ≥ 0. The two sequences will be shown to behave like S_n5 and S_n13 of Part XIXB where one of their equivalents is based on the square root of a cubed expression.

All the terms above (the D values) were entered into the Pell calculator program to give values of Q_n = +1 for different values of n. It should be noted that the x and the y terms are found at the bottom of column n − 1 just to the left of Q_n in the coded examples. We can see that in the two coded examples: for 717 at Code 717 and for 741 Code 741. The red number −3 although generated by the equation in sequence S_n-3 being negative is not usable. The two numbers 21 and 69 have their Q_n = 1 at n = 7 and 9, respectively (see Code 21 and Code 69), which differ from Q_n = 1 at n = 11 and 13, respectively for the rest of the Ds in S_n21 and S_n-3.

Tables of x, y and D values

Tables I and II show the first 11 D numbers in the two different sequences S_n. Although the values of D equal 69 and 21 in Tables I and II, respectively, they are tabulated even though they do not have the requisite Q_n = 1 at n = 13 and 11, respectively, shared with the other D terms in the equations.

where the values of y for S_n-3 and S_n21 are the consecutive odd numbers and D and y are divisible by three.

Tables of Pell Equations

The Pell equations are set up according to the following format where R_D is equal to the equivalent expressions and where n₁ is a constant multiplied by a consecutive triangular number and (2n+1) is a consecutive odd number:

where n ≥ 2 for all but 21 and 69.

Tables I and II give the formulas for all 10 Pell equations according to the format listed above with the triangular and odd numbers in blue. Note that the values of n₁ are off for R₂₁ and R₆₉.

This concludes Part XIXC. To go back to Part XIXB.

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