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

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:

x = [(p + q√D)ⁿ + (p − q√D)ⁿ ∕ 2]
y = [(p + q√D)ⁿ + (p − q√D)ⁿ ∕ 2√D)]

It was shown in Part XV that the sequence S_m having the OEIS number A013643 was found to consist of two sequences one of which is the equation (5n + 1)² + 4n + 1 and the other some random sequence of numbers.

That sequence:

41, 130, 269, 370, 458, 697, 986, 1313, 1325, 1613, 1714, 2153, 2642, 2834, 3181, 3770, 4409, 4778, 4933, 5098, 5837, 5954, ...

where the numbers in red correspond to the equation (5n+1)² + 4n + 1 and the ones in black appear to correspond to the random part. If we look at the continued fraction expansion the first is [5n+1,2,2,10n+2]. The random part is not consistent but appear to be different for each black number. This can be seen in a_n (row 4, columns 2-5) of the four computer examples: Code 41, Code 130, Code 1313 and Code 1613.

A new squence has been found consisting of two sequences, the red one above and a new one having a similar type of generating equation:

2, 13, 41, 74, 130, 185, 269, 346, 458, 557, 697, 818, 986, 1129, 1325, 1490, 1714, 1901, 2153, 2362...

where the value of y is constant and the n of the Q_n = 1 values alternate between 4 and 6. The equation of the red sequence is again (5n + 1)² + 4n + 1 and that of the black sequence as (5n + 3)² + 6n + 4, where the 5n + 1 and 5n + 3 correspond to the first number in the continued fraction expansion. The continued fraction expansion for the black sequence is [5n+3,1,1,1,1,10n+6] much different from that of the red sequence yet similar as seen in Table I. To see two examples from that sequence Code 74 and Code 557.

Utilizing the computer program in Part XIII we can rapidly generate the data tabulated in Table I for the D, x and y values along with the n position where Q_n = 1. Moreover, the data fulfill the negative Pell equation x² − Dy² = −1. The regular Pell (Table II) can be found one column to the left of n=7 for the red and one column to the left of n=11 for the black sequences, respectfully.

Table I (−Pell)
D	2	13	41	74	130	185	269	346	458	557	697	818
x	7	18	32	43	57	68	82	93	107	118	132	143
y	5	5	5	5	5	5	5	5	5	5	5	5
Q_n=1	4	6	4	6	4	6	4	6	4	6	4	6

Table II (+Pell)
D	2	13	41	74	130	185	269	346	458	557	697	818
x	99	649	2049	3699	6499	9249	13449	17299	22899	27849	34849	40899
y	70	180	320	430	570	680	820	930	1070	1180	1320	1430
Q_n=1	7	11	7	11	7	11	7	11	7	11	7	11

where the differences between adjacent ys in Table II alternate between 110 and 140.

This concludes Part XXI. Go to Part XV.

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

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