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

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,12,1), where x equals to 55, 73 or 89. In addition, the triple (55,12,1), (73,12,1) and (89,12,1) belongs to D values of 21, 37 and 55, respectively in the articles listed above. It will be shown here that we can continue generating all those values of x which may or may not be listed in these articles by employing what appears to be a new sequence but in reality is a mixture of two sequences:

2, 21, 35, 37, 55, 112, 142, 146, 180, 275, 321, 327, 377, 510, 572, 580, 646, 817, 895, 905, 987, 1196, 1290, 1302, 1400, 1647, 1757, 1771,...

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 = 2
P(n) =
F_2n+1 = F_2n + 19(2n-1) F_2n+2 = F_2n+1 + 34n
P(m) = (m(36m − 1)), (m(36m + 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 = 2 and the counter n set to 1. Thus, according to the second line F₃ = 21 and F₄ = 55 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
19 is the difference between 2 and 21 and 34 is the difference between 21 and 55
Each pair of P(n) and P(m) employs the same respective n and m

It is noted here that these dual sequences are a property displayed by many numbers, where fifteen of the following y values associated with these new sequences fall under 51:

12, 15, 20, 21, 24, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48

Besides the D=12 on this page, the 24 Part XB, the 15 Part XC, the 20 Part XD and the 21 Part XE are available and the equations for the dual sequences have been found to be slightly different for all five Ds.

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 12.

Table D
n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
D(n)	2	21	55	112	180	275	377	510	646	817	987	1196	1400	1447
x	17	55	89	127	161	199	233	271	305	343	377	415	449	487

D(m)	35	37	142	146	321	327	572	580	895	905	1290	1302	1757	1771
x	71	73	143	145	215	217	287	289	359	361	431	433	503	505

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 ∕ ⅓n₁ = x + 12√D
R(m)_D = (6m₁ + √D)² ∕ ⅙m₁ = x + 12√D

where:

n₁ are the odd integers: 6n + 3 starting at n = 0
the values of adjacent xs in a pair is (36(2n − 1) − 19), (36(2n − 1) + 19)
D are the values from the above sequence starting at 2.

m₁ are consecutive integers: 1, 2, 3, 4, 5, ...
the values of adjacent xs in a pair is (72m₁ − 1), (72m₁ + 1)
D are the values from the above sequence starting at 35.

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.

Before looking at the examples it must not be overlooked that the initial numbers of each of the sequences are not the first least solutions to the Pell equation but are, in this case, the second. So one must be careful not to attribute these values to new least solutions. Other sequences (see Parts XB thru XE) may also show this.

Tables of D and Pell (x,12,1) Triples

Note: All the (n₁ + √D)² ∕ ⅓n₁ and (6m₁ + √D)² ∕ ⅙m₁ 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 − 2y² = 1 using R₂ = (3 + 2√2)² = 17 + 12√2 = 33.97056275
Table II shows the triples for the Pell equation x² − 21y² = 1 using R₂₁ = (9 + 2√21)² ∕3 = 55 + 12√21 = 109.9909083

Table I D=2
x	y	z
17	12	1
577	408	1
19601	13860	1

Table II D=21
x	y	z
55	12	1
6049	1320	1
665335	145188	1

Table III shows the triples for the Pell equation x² − 35y² = 1 using R₃₅ = (6 + √35)² = 71 + 12√35 = 141.9929574
Table IV shows the triples for the Pell equation x² − 37y² = 1 using R₃₇ = (6 + √37)² = 73 + 12√37 = 145.9931504

Table III D=35
x	y	z
71	12	1
10081	1704	1
1431431	241956	1

Table IV D=37
x	y	z
73	12	1
10657	1752	1
1555849	255780	1

Table V shows the triples for the Pell equation x² − 55y² = 1 using R₅₅ = (15 + 2√55)² ∕5 = 89 + 12√55 = 177.9943818
Table VI shows the triples for the Pell equation x² − 112y² = 1 using R₁₁₂ = (21 + 2√112)² ∕7 = 127 + 12√112 = 253.9960629

Table V D=55
x	y	z
89	12	1
15841	2136	1
2819609	380196	1

Table VI D=112
x	y	z
127	12	1
32257	3048	1
8193151	774180	1

Table VII shows the triples for the Pell equation x² − 142y² = 1 using R₁₄₂ = (12 + √142)² ∕2 = 143 + 12√142 = 285.9965035
Table VIII shows the triples for the Pell equation x² − 146y² = 1 using R₁₄₆ = (12 + √146)² ∕2 = 145 + 12√146 = 289.9965577

Table VII D=142
x	y	z
143	12	1
40897	3432	1
11696399	981540	1

Table VIII D=146
x	y	z
145	12	1
42049	3480	1
12194065	1009188	1

Table IX shows the triples for the Pell equation x² − 180y² = 1 using R₁₈₀ = (27 + 2√180)² ∕9 = 161 + 12√180 = 321.9968944
Table X shows the triples for the Pell equation x² − 275y² = 1 using R₂₇₅ = (33 + 2√275)² ∕11 = 199 + 12√275 = 397.9774874

Table IX D=180
x	y	z
161	12	1
51841	3864	1
16692641	1244196	1

Table X D=275
x	y	z
199	12	1
79201	4776	1
31521799	1900836	1

Table XI shows the triples for the Pell equation x² − 321y² = 1 using R₃₂₁ = (18 + √321)² ∕3 = 215 + 12√321 = 429.9976744
Table XII shows the triples for the Pell equation x² − 327y² = 1 using R₃₂₇ = (18 + √327)² ∕3 = 217 + 12√327 = 4.339976958

Table XI D=321
x	y	z
215	12	1
92449	5160	1
39752855	2218788	1

Table XII D=327
x	y	z
217	12	1
94177	5208	1
40872601	2260260	1

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

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

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

Tables of D and Pell (x,12,1) Triples

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