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

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 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√2D)ⁿ ∕ 2√2D)]

The tables in these two articles show a series of numbers of which I will focus on those triples of the type (x,24,1), where x equals to 127 or 161. In addition, the triple (127,24,1) and (161,24,1) belongs to D values of 28 and 45, respectively, in the wiki article listed above. 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, 28, 45, 143, 145, 299, 350, 574, 578, 858, 943, 1293, 1299, 1705, 1824, 2300, 2308, 2840, 2993, 3595, 3605, 4263, 4450...

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 = 28
P(n) =
F_2n+1 = F_2n + 17(2n-1) F_2n+2 = F_2n+1 + 254n
P(m) = (m(144m − 1)), (m(144m + 1))

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 = 28 and the counter n set to 1. Thus, according to the second line F₃ = 45 and F₄ = 299 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
17 is the difference between 28 and 45 and 254 is the difference between 45 and 299
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 24.

Table D
n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
D(n)	28	45	299	350	858	943	1705	1824	2840	2993	4263	4450	5974	6195
x	127	161	415	449	703	737	991	1025	1279	1313	1567	1601	1855	1889

D(m)	143	145	574	578	1293	1299	2300	2308	3595	3605	5178	5190	7049	7063
x	287	289	575	577	863	865	1151	1153	1439	1441	1727	1729	2015	2017

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 + 2√D)² − 21] ∕ ⅙n₁ = x + 24√D
R(m)_D = (12m₁ + √D)² ∕ ¹⁄₁₂m₁ = x + 24√D

where:

n₁ are the even integers: 12n + 6 starting at n = 0
the values of adjacent xs in a pair is (144(2n − 1) − 17), (144(2n − 1) + 17)
D are the values from the above sequence starting at 28.

m₁ are consecutive integers: 1, 2, 3, 4, 5, ...
the values of adjacent xs in a pair is (288m₁ − 1), (288m₁ + 1)
D are the values from the above sequence starting at 143.

then multiplying and rounding off each row of triples generated by the R_D for as many triples as are desired. The lists below show the patterns generated for twelve Ds of the above sequence and the accompanying triple tables generated for each D where a D may be even number or odd.

Tables of D and Pell (x,24,1) Triples

Note: All the [(n₁ + √D)² − 21] ∕ ⅙n₁ and (12m₁ + √D)² ∕ ¹⁄₁₂m₁ are present in pairs. In addition, a D(n) pair is shown first followed by a D(m) pair.
Table I shows the triples for the Pell equation x² − 28y² = 1 using R₂₈ = (6 + 2√28)² − 21 = 127 + 24√28 = 253.9960629
Table II shows the triples for the Pell equation x² − 45y² = 1 using R₄₅ = [(18 + 2√45)² − 21] ∕3 = 161 + 24√45 = 321.9968944

Table I D=28
x	y	z
127	24	1
32257	6096	1
8193151	1548360	1

Table II D=45
x	y	z
161	24	1
51841	7728	1
16692641	2488392	1

Table III shows the triples for the Pell equation x² − 143y² = 1 using R₁₄₃ = (12 + √143)² = 287 + 24√143 = 573.9982578
Table IV shows the triples for the Pell equation x² − 145y² = 1 using R₁₄₅ = (12 + √145)² = 289 + 24√145 = 577.9982699

Table III D=143
x	y	z
287	24	1
164737	13776	1
94558751	7907400	1

Table IV D=145
x	y	z
289	24	1
167041	13872	1
96549409	8017992	1

Table V shows the triples for the Pell equation x² − 299y² = 1 using R₂₉₉ = [(30 + 2√299)² − 21] ∕5 = 415 + 24√299 = 829.9987952
Table VI shows the triples for the Pell equation x² − 350y² = 1 using R₃₅₀ = [(42 + 2√350)² − 21] ∕7 = 449 + 24√350 = 897.9988864

Table V D=299
x	y	z
415	24	1
344449	19920	1
285892255	16533576	1

Table VI D=350
x	y	z
449	24	1
403201	21552	1
362074049	19353672	1

Table VII shows the triples for the Pell equation x² − 574y² = 1 using R₅₇₄ = (24 + √574)² ∕2 = 575 + 24√574 = 1149.999913
Table VIII shows the triples for the Pell equation x² − 578y² = 1 using R₅₇₈ = (24 + √578)² ∕2 = 577 + 24√578 = 1153.999133

Table VII D=574
x	y	z
575	24	1
661249	27600	1
760435775	31739976	1

Table VIII D=578
x	y	z
577	24	1
665857	27696	1
768398401	31961160	1

Table IX shows the triples for the Pell equation x² − 858y² = 1 using R₈₅₈ = [(54 + 2√858)² − 21] ∕9 = 703 + 24√858 = 1405.999289
Table X shows the triples for the Pell equation x² − 943y² = 1 using R₉₄₃ = [(66 + 2√943)² − 21] ∕11 = 737 + 24√943 = 1473.999322

Table IX D=858
x	y	z
703	24	1
988417	33744	1
1389713599	47444040	1

Table X D=943
x	y	z
737	24	1
1086337	35376	1
1601260001	52144200	1

Table XI shows the triples for the Pell equation x² − 1293y² = 1 using R₁₂₉₃ = (36 + √1293)² ∕3 = 863 + 24√1293 = 1725.999421
Table XII shows the triples for the Pell equation x² − 1299y² = 1 using R₁₂₉₉ = (36 + √1299)² ∕3 = 865 + 24√1299 = 1729.999422

Table XI D=1293
x	y	z
863	24	1
1489537	41424	1
2570939999	71497800	1

Table XII D=1299
x	y	z
865	24	1
1496449	41520	1
2588855905	71829576	1

This concludes Part XB. Go to Part XC. Go back to Part XA.

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

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

Tables of D and Pell (x,24,1) Triples

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