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

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

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:

Although the tables in these two articles show a series of Ds, x, and ys used in the Pell equation I will not focus on any of these since they are not in the range of values required. Instead I will focus on those triples of the type (x,y,−1) using the negative Pell equation where we initially have two Ds, 373 and 533, which have an identical y value of 265. Before I turn to this topic I'll first post the first 20 values of a new sequence S_n not listed in the OEIS database, corresponding to the various D values with y = 265 and I'll be employing the negative Pell equation x² − Dy² = −1 to determine the x values of D equal to 373 and 533. The sequence S_n:

where each group of two numbers have been separated into pairs since they are related by a certain property. The black and red pairs are different enough that both belong to two different sequences, S_n1 and S_n2, respectively.

There are a couple of ways to do this. First according to the literature both D values of 373 and 533 have large values of y for the positive Pell equation, where the equation is equal to 1. The y and x values obtained from Google's Canon Pellianus are plugged into the equation x + y√D:

followed by taking the square roots of the R values:

where a specific R_D equals a specific x + y√D. Dividing the two large numbers in each of the equations by the appropriate R^½ gives:

Thus we have converted the + Pell numbers into their − Pell counterparts. The alternative approach, however, is to use the computer program (A Pell calculator) developed on this website (see Calculator) to find these values. Moreover, any D can be typed in. Easy peasy. For example, the calculations for 373 and 533 are shown in Calc 373 and Calc 533. In addition, the negative Pell numbers are the way to go since the positive Pell numbers are too cumbersome to work with.

And now to the juicy part the generation of the Sequence S_n. We set up a table where the first row consists of the number (Δ) to add to each number to generate the next number in the sequences of x that number being 70225 the square of y = 265. The third column contains the two xs 5118 and 6118, respectively. Subtracting 70225 from these two numbers gives −65107 and −64107 which are the required two numbers after 5118 and 6118 (after converting to positive numbers). It doesn't matter whether x is + or − in the Pell equation since x is being squared. These two numbers go into the last two rows and are subjected to the same addition with 70225. Since the same number is being added each time we can reduce this to a general equation where we can consider the numbers at n = 0, column 3 as being the initial x₀ and use the equation:

where the general equation for the negative and positive Pell is:

for all n ≥ 0.

Note that the two pair of xs differ by 1000. Strange but true. Using every one of these calculated xs and the lone y = 265 along with the negative Pell equation we can calculate for all Ds and obtain the results shown in Table II. However, there are two alternative ways to obtain the D values. We can use the equation for obtaining the D values by using the recursive formula:

for all n ≥ 0.

Or we can modify the Pellian formula and substitute the value of x in (a) to get a random access instead of the sequential equation above:

and since

from above in the R_D section, then

and

where (a) and (c) are the desired equations for generating all xs and all Ds for the sequence S_n. We can generalize these equations by using (b) and (d) as long as the initial variables are x₀ and D₀.

for all n ≥ 0 in terms of D₀ and y.

Similarly the next equation can be used to generate the positive Pell solutions if one desires to go via that route instead:

Thus, one can see from scanning down the columns in Table II, that the numbers are identical to the original sequence S_n.

I mentioned above that the entries in the two sequences are related by a certain property. The method of using continuing fractions as used in the Pell calculator relies on the fact that when Q_n = 1 in column n of the Pell calculator, the x and y values that satisfy the Pell equation fall at n − 1. For the negative Pell the value of n − 1 is an odd number, while the value for a positive Pell n − 1 is an even number.

Q_n = 1 for the first subsequence S_n1 falls at an n = 6 with a period of 5 and for the second subsequence S_n2 falls at n = 8 with a period of 7. This is shown for 58522 and 60362 at Calc 58522 and Calc 60362. So these two sequences although having the same y value are not identical.

This concludes Part XVI. Go back to Part XV.

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