Consecutive Odd Numbers and the Pellian Equation(Part XIXB)

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 XIXA, 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 below and two of the sequences, 36(n)(n+1) + 13 and 36(n)(n+1) + 5 were found to be subsets of the equations ending above in 5 and −3, respectively:

where all the sequences were evaluated at n ≥ 0. No values exist for S_n9 = 36(n)(n+1) + 9 (dashed line), since the terms generated are perfect square integers which are not allowed as values of D in the Pell equation and, consequently, when used in the program to calculate convergents will result in a "no integer solution".

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 six coded examples: for 723 at Code 723; for 725 Code 725; for 727 at Code 727; for 1091 at Code 1091; for 1093 Code 1093; and for 1095 Code 1095. The red numbers 3 and 5 although generated by the equation in sequence S_n3 do not share the properties of the numbers under discussion since their Q_n = 1 at n = 3,5,7,... for D = 3 and n = 1,1,1,... for D = 5 (see Code 3 and Code 5).

Tables of x, y and D values

Tables I-VI show the first 11 D numbers in the six different sequences S_n. Although the values of D equal 3 and 5 in Tables I and II, respectively, they are tabulated even though they do not have the requisite properties, i.e., Q_n = 1 shared with the other D terms of the equations.

where the values of y for S_n3, S_n7, S_n11 and S_n15 and the values of x in S_n13 are the consecutive odd numbers. For S_n5, on the other hand, y is now a non consecutive triangular number ½[(3m + 1)² + (3m + 1)]. Lastly, from the tables we can see that x takes on the following values as symmetrical ± pairs:

Tables of Pell Equations

The Pell equations are set up according to the following formats with the x and ys taken from the above six tables. The first expression with n₁ gives the alternate expression for x + y√D where the square term is divided by either 2 or 6 and n₁ is a consecutive odd number = 3(2n + 1). The second expression with n₂ gives the alternate expression x + y√D for S_n5 and S_n13 where n₂ is a consecutive odd number × 3. Again in four out of six cases y is a consecutive odd number, in another it is a triangular number, and in the last case it is x which is the odd consecutive number.

Tables VII-IX give the formulas for all 10 Pell equations according to the format listed above with the triangular and odd numbers in blue.

This concludes Part XIXB. To go to the next continuation Part XIXC.
To go back to Part XIXA.

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