The Pellian Equation x² −Dy² = ±1 from Dual Sequences (Part XV)

Ramdoness Derandomized

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:

It was shown in the Part XIV that the sequence S_m having the OEIS number A002522 containing the equation (n + 1)² + 1 or n² + 1, as the case may be, is the continued fraction expansion of sqrt(n) having a period of 1 and thus can produce a subset of the negative Pell values. It will be shown that these numbers play a special part in this article.

In addition, the following sequence from the OEIS database A013643 contains a number of values correponding to the D values with a continued fraction having a period of 3:

where the numbers in red correspond to the equation (5n+1)² + 4n + 1 and the ones in black appear to correspond to no equation at all but appear to be random. This can be visualized in Table I where the bold numbers do not follow a consistently sequential pattern as the other group of numbers do. What it does show is that the Q_n values of these numbers, viz., 370, 1313 and 1613 have the same Q_n values as the D values of 130, 458 and 269, respectively.

All the other black numbers in the original OEIS sequence if allowed to go on indefinitely behave similarly and don't follow any specific order as the red numbers do. However, looks can be deceiving.

Order Out of Chaos - A Dual Method

Method I

So let's make sense of all this. This is the original method I came up with. A second new method just found will be described below. The thing is to collect all the black numbers (an infinite quest) so instead we take the first few and find the values of x and y in the equation x + y√D . These can be found in a table of values for example in the Google Canon Pellianus which gives values up to 1000. Not good for the work done here so instead we use the computer program (A Pell calculator) developed on this website shown in Part XIII. The Q_n values listed for each D in Table I were found in this way.

For example, a JS calculation of the values of x and y shows that for D = 1313 for the negative Pell (where the equation equals −1), x and y are 17 and 616, respectively, at n=3. The values at n=6 are also important since they are the least solutions for the positive Pell equation where the equation equals 1. However, these numbers become larger as D increases, viz., x=758913 and y=20944. So only the negative Pell values will be considered.

Now that we have obtained one set of x and y values we do the same for the other black numbers in the sequence and we keep all the D values of one type together in a table (see Table II). The second, third and fourth entries obtained from the sequence above all have y=17. It can be seen that the difference (Δ) between each adjacent x is equal to 289 which is the square of y which equals 17. Adding 289 to the fourth number gives 1483. And adding 289 to this gives 1772. Thus we have a means of collecting all the like D values, generating order and producing an infinitude of numbers.

Now for the red entry. The first number 370 is larger than 289 so if we do a subtraction we get x to be 38 and since we know y=17 we can calculate D using the negative Pell equation. Checking the D value of 5 in the Pell calculator we see that the Q_n values are equal to 1 for every n and that the x,y values we desire are at n=3. A good confirmation.

Thus we have found that a Q_n of a period of 1 can generate a Q_n of a period of 3.

To continue what are we to do with the value of 1613. Using a D value of 1613 in the Pell calculator we see that the values obtained for x and y can be similarly placed in Table III and the same procedure applied using Δ=37²=1369 to generate all equations having the same D values including the initial equation in red. Table III shows that the third entry is the last value in the sequence above.

With the results we have obtained thus far we have a foothold on how to proceed. We have two sets of numbers and, therefore, two sets of equations which we can use to generate Table IV, where the top row of Ds shows a relationship with the bottom row of ys. These two equations n² + 1 (with the OEIS Sloane number A002522) and 4n² + 1 (with the OEIS Sloane number A053755) are then useful for contruction of the appropriate tables V, VI and VII. This is where the equation of the previous article Part XIV becomes useful.

Table V contains as a check the D value 4778 from the sequence above. Table V and VI also contain the respective values 17989 and 11257 shown in the Sloane number A013643 for that sequence. Since A013643 only contains 39 entries it is not possible to check out any of the other numbers generated here. However, the numbers generated by this article could easily overwhelm this particular Sloane entry. For example, a D value over a million is produced for the sixth value when y=257:   332317 + 257√1672010.

Method II - A Less Confusing Alternative Approach

The second method (which I wish I would have found sooner) uses the pair of equations listed below in (a) and (b) where x₀ and D₀ correspond to the initial values that doesn't require the use of any of the Sloane sequences listed above.

for all n ≥ 0.

Thus, using the initial values from Table V for x₀, y₀ and D₀ of 4493, 65 and 4778, respectively we plug them into the two equations above derived in Part XVI and by varying n we get all the values listed in Table V without the first row values in red. Too bad I didn't think of it earlier. It's Occam's razor applied to Pell. Simple and less confusing.

One more thing, the two equations can be used to find all subsequent red D_ns after the D₀ value of 41 in the original sequence S_n. Thus, this method is complementary to the original Sloane number A007533 consisting of the formula (5n+1)² + 4n + 1.

This concludes Part XV. Go forward to Part XVI. Go back to Part XIV.

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