The Role of Triangular Numbers in the Pellian Equation x² −Dy² = 1 (Part XVIII)

A quick summary of the Pellian equation x² − Dy² = 1 is covered in Part I and the negative Pellian equation x² − Dy² = −1 in Part II.

On the previous article Part XVII, the following equation and its accompanying sequence S_n, were shown to correspond to those D values of the Pell equation whose Q_n was equal to +1 at n = 7. (See article on Convergents Part XIII for a description of Q_n):

These D terms when plugged into a Pell equation generated the appropriate x and y variables consistent with their D values. In addition, the equations could be written in an alternative mode in which the y values (as the value n₂) were triangular numbers ½(n²+n). See Part XVII Table II. Moreover, I will show that the equivalent form of the x variables, as depicted in Part XVII, are the sum of triangular numbers and a pair of numbers from a known OEIS sequence.

Triangular Numbers to x Conversion

The Sloane sequence A002620 was found to be identical to a sequence I generated from the pair of values (m², m²+m) where m ≥ 0. This pair of m values were, however, not the end-all but needed a doubling of each of the terms in the sequence, as in the third row of the accompanying table, prior to using them in their conversion to x:

Note that the sequence is constructed in such a manner that the difference between the pair of m values on the second row is also a pair of Δ values.

We begin by employing a second variable n to generate the triangular numbers ½(n²+n) which when subjected to multiplication followed by addition generate x:

where k = 2m² or 2(m²+m) depending on whether column 3 or 4 of Table I is being used and where the latter number is twice a triangular number. Column 2 contains the values of m which coincidentally are identical to Δ. The last column are the values of y identical to the triangular numbers that were initially shown in Part XVII Table II along with their accompanying D and x values. There one uses D to find x and y. Here one can do the reverse and use x and y to find D if one is so inclined.

Thus, we have the first instance of a series of equations equivalent to their Pell counterparts where their x and y values are based on the triangular numbers.

This concludes Part XVIII. To Go back to Part XVII.

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