Consecutive Odd Numbers and the Pellian Equation(Part XIXA)

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 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 for 4(n)(n+1) + 5 in Part IIIA and for 4(n)(n+3) + 5 in Part XVII. Two other equations have been found in the OEIS database S_n41 = 4(n)(n+1) − 1, A073577 and S_n43 = 4(n)(n+1) + 3, A1648971, respectively, are listed below:

where n ≥ 0. Six other new sequences of the type 36(n)(n+1) + m have been constructed (not listed in the OEIS) and are the topic of the next section Part XIXB Continuation.

All the terms above (the D values) were entered into the Pell calculator program which gave the values of Q_n = +1 at n = 3,5,7, etc. for each term in the first sequence and n = 5,9,13, etc. for the terms in the second sequence. 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 (one term from each of the two sequences): for 123 at Code 123; for 223 at Code 223. The red numbers 3 and −1 in the first and second sequences although generated by the equation do not share the properties of the numbers under discussion since their Q_n is either 1 at a different n (see Code 3) or non existent for negative D.

Tables I and II show the first 11 D numbers in the two different sequences S_n. Only the latter 10 terms in S_n2 and S_n3 have the requisite properties not shared with the first term, viz., the Q_n are the same for each group.

In addition, the values of y are the consecutive odd numbers while x takes on the values:

The Pell equations are set up according to the following format with the x and ys taken from the above two tables. The expression with n₁ gives the alternate expression for x + y√D where the square term is divided by 2 and n₁ is a consecutive odd number = 2n + 1. Again in all cases y is a consecutive odd number.

Table III gives the formulas for all 10 Pell equations according to the format listed above. Both left and right equations contain a consecutive odd number (in blue) in both the n₁ and y variables.

This concludes Part XIX. Go to Part XIXB Continuation. To go back to Part XVII.

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