The Pellian Equation x² −Dy² = 1 from two Paired Sequences P(n)/P(m) (Part XE)

A 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, in Google books:Canon Pellianus with about 1000 entries and on 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√2D)ⁿ ∕ 2√2D)]

The tables in the Canon Pellianus article shows a list of numbers corresponding to triples of the type (x,21,1), where seven D values (88-923) and their corresponding x and y values. It will be shown here, as was shown in Part XA for y = 12, that we can continue generating all those values of x not listed in these articles by employing what appears to be a new sequence but in reality is a mixture of two sequences:

0, 0, 88, 135, 439, 443, 923, 1064, 1760, 1768, 2640, 2875, 3963, 3975, 5239, 5568, 7048, 7064, 8729, 9143, 11015, 11035,...

Since one equation cannot capture all the numbers in the sequence the single sequence can be split into two different paired sequences composed of the following two expressions:

F₂ = 88
P(n) =
F_2n+1 = F_2n + 47(2n-1) F_2n+2 = F_2n+1 + 788n
P(m) = (m(441m − 2)), (m(441m + 2))

where the first expression P(n) is composed of a pair of numbers, each number is the sum of the preceding one starting out with an initial value, F₂ = 88 and the counter n set to 1. Thus, according to the second line F₃ = 135 and F₄ = 923 with F₄ subsequently used in the next line when n is incremented to 2. The initial pair is consequently (F₂,F₃) followed by (F₄,F₅) followed by (F₆,F₇), etc. consistent with the sequence P(n). As for the second expression, P(m) is treated as a paired sequence which uses a pair of equations to generate the two paired values.

The other properties of these sequences are:

n and m are both initially set to 1 and incremented by 1
47 is the difference between 88 and 135, and 788 is the difference between 135 and 923
Each pair of P(n) and P(m) employs the same respective n and m

Table D shows the various Ds from the two split sequences P(n) and P(m) along with their respective x values. All y values are 21.

Table D
n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
D(n)	88	135	923	1064	2640	2875	5239	5568	8720	9143	13083	13600	18328	18939
x	197	244	638	685	1079	1126	1520	1567	1961	2008	2402	2449	2843	2890

D(m)	439	443	1760	1768	3963	3975	7048	7064	11015	11035	15864	15888	21595	21623
x	440	442	881	883	1322	1324	1763	1765	2204	2205	2645	2647	3086	3088

Both P(n) and P(m) use the same method but the mathematical expressions are different and involves multiplying the initial least solutions by either of the two parts of the following mathematical expression:

R(n)_D = (n₁ + 3√D)² ∕ ¹⁄_3.5n₁ = x + 15√D
R(m)_D = (21m₁ + √D)² ∕ ¹⁄_10.5m₁ = x + 15√D

where:

n₁ are the integers: 7(9n + 4) and 7(9n + 5) starting at n = 0 for the pair
the values of adjacent xs in a pair is (441(2n − 1) − 47)/2), (441(2n − 1) + 47)/2)
D are the values from the above sequence starting at 88.

m₁ are consecutive integers: 1, 2, 3, 4, 5, ...
the values of adjacent xs in a pair is (441m₁ − 1), (441m₁ + 1)
D are the values from the above sequence starting at 439.

The first three pairs of D(n) and D(m) from Table D and their corresponding Equal Expressions are tabulated in Table I.

Pell Equal Expressions for (x,21,1) Triples

Note: All the (n₁ + √D)² ∕ ¹⁄_3.5n₁ and (21m₁ + √D)² ∕ ¹⁄_10.5m₁ are present in pairs. In addition, a D(n) pair is shown first followed by a D(m) pair.

Table I
Pell Equation	Equal Expressions
x² − 88y² = 1	R₈₈ = (28 + 3√88)² ∕8 = 197 + 21√88
x² − 135y² = 1	R₁₃₅ = (35 + 3√135)² ∕10 = 244 + 21√135
x² − 439y² = 1	R₄₃₉ = (21 + √439)² ∕2 = 440 + 21√439
x² − 443y² = 1	R₄₄₃ = (21 + √443)² ∕2 = 442 + 21√443
x² − 923y² = 1	R₉₂₃ = (91 + 3√923)² ∕26 = 638 + 21√923
x² − 1064y² = 1	R₁₀₆₄ = (98 + 3√1064)² ∕28 = 685 + 21√1064
x² − 1760y² = 1	R₁₇₆₀ = (42 + √1760)² ∕4 = 881 + 21√1760
x² − 1768y² = 1	R₁₇₆₈ = (42 + √1768)² ∕4 = 883 + 21√1768
x² − 2640y² = 1	R₂₆₄₀ = (154 + 3√2640)² ∕44 = 1079 + 21√2640
x² − 2875y² = 1	R₂₈₇₅ = (161 + 3√2875)² ∕52 = 1126 + 21√2875
x² − 3963y² = 1	R₃₉₆₃ = (63 + √3963)² ∕6 = 1322 + 21√3963
x² − 3975y² = 1	R₃₉₇₅ = (63 + √3975)² ∕6 = 1324 + 21√3975

This concludes Part XE. Go to Part XFa. Go back to Part XD.

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The Pellian Equation x2 −Dy2 = 1 from two Paired Sequences P(n)/P(m) (Part XE)

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

Pell Equal Expressions for (x,21,1) Triples

The Pellian Equation x² −Dy² = 1 from two Paired Sequences P(n)/P(m) (Part XE)