The Pellian Equation x² −Dy² = 1 from a Paired Sequence P(n) (Part VIII)

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. 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,3,1). x equal to 8 or 10 was described in Part I. In addition, the triples (8,3,1) and (10,3,1) belong to D values of 7 and 11, respectively, in the articles listed above. It will be shown here that we can continue generating all those values of x not listed in these articles by employing the sequence with the OEIS number A132355 which can be generated using the equation for a pair of numbers P(n) = n(9n − 2), n(9n + 2) corresponding to the various Ds:

0, 0, 7, 11, 32, 40, 75, 87, 136, 152, 215, 235, 312, 336, 427, 455, 560, 592, 711, 747, 880, 920, ...

where an extra zero has been added to the sequence so that P(0) = (0,0). In addition, each pair uses the same value of n.

The method involves multiplying the initial least solutions by either of the two parts of the following mathematical expression:

R_D = (3n₁ + √D)² ∕ 2n₁ = x + 3√D)

where:

n₁ are consecutive integers, 1, 2, 3, 4, 5, ...
the values of adjacent xs in a pair is (9n₁ − 1, 9n₁ + 1).
D are the values from the above sequence starting at 7.

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 the ten 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,3,1) Triples

Note: all the (n₁ + √D)² ∕ ½n₁ are present in pairs.
Table I shows the triples for the Pell equation x² − 7y² = 1 using R₇ = (3 + √7)² ∕ 2 = 8 + 3√7 = 15.93725393
Table II shows the triples for the Pell equation x² − 11y² = 1 using R₁₁ = (3 + √11)² ∕ 2 = 10 + 3√11 = 19.94987437
Table III shows the triples for the Pell equation x² − 32y² = 1 using R₃₂ = (6 + √32)² ∕ 4 = 17 + 3√32 = 33.97056275

Table I D=7
x	y	z
8	3	1
127	48	1
2024	765	1
32257	12192	1
514088	194307	1

Table II D=11
x	y	z
10	3	1
199	60	1
3970	1197	1
79201	23880	1
1580050	476403	1

Table III D=32
x	y	z
17	3	1
577	102	1
19601	3465	1
665857	117708	1
22619537	3998607	1

Table IV shows the triples for the Pell equation x² − 40y² = 1 using R₄₀ = (6 + √40)² ∕ 4 = 19 + 3√40 = 37.97366596
Table V shows the triples for the Pell equation x² − 75y² = 1 using R₇₅ = (9 + √75)² ∕ 6 = 26 + 3√75 = 51.98076211
Table VI shows the triples for the Pell equation x² − 87y² = 1 using R₈₇ = (9 + √87)² ∕ 6 = 28 + 3√87 = 55.98213716

Table IV D=40
x	y	z
19	3	1
721	114	1
27379	4329	1
1939681	164388	1
39480499	6242415	1

Table V D=75
x	y	z
26	3	1
1351	156	1
70226	8109	1
3650401	421512	1
189750626	21910515	1

Table VI D=87
x	y	z
28	3	1
1567	168	1
87724	9405	1
4910977	526512	1
274936988	29475267	1

Table VII shows the triples for the Pell equation x² − 136y² = 1 using R₁₃₆ = (12 + √136)² ∕ 8 = 35 + 3√136 = 69.98571137
Table VIII shows the triples for the Pell equation x² − 152y² = 1 using R₁₅₂ = (12 + √152)² ∕ 8 = 37 + 3√152 = 73.98648402
Table IX shows the triples for the Pell equation x² − 215y² = 1 using R₂₁₅ = (15 + √215)² ∕ 10 = 44 + 3√215 = 87.9886349

Table VII D=136
x	y	z
35	3	1
2449	210	1
171395	14697	1
11995201	1028580	1
839492675	71985903	1

Table VIII D=152
x	y	z
37	3	1
2237	222	1
202501	16425	1
14982337	1215228	1
1108490437	89910447	1

Table IX D=215
x	y	z
44	3	1
3871	264	1
340604	23229	1
29969281	2043888	1
2636956124	179838915	1

Table X shows the triples for the Pell equation x² − 235y² = 1 using R₂₃₅ = (15 + √235)² ∕ 10 = 46 + 3√235 = 91.98912915
Table XI shows the triples for the Pell equation x² − 312y² = 1 using R₃₁₂ = (18 + √312)² ∕ 12 = 53 + 3√312 = 105.9905652
Table XII shows the triples for the Pell equation x² − 336y² = 1 using R₃₃₆ = (18 + √336)² ∕ 12 = 55 + 3√336 = 109.9909083

Table X D=235
x	y	z
46	3	1
4231	276	1
389206	25389	1
35802721	2335512	1
3293461126	214841715	1

Table XI D=312
x	y	z
53	3	1
5617	318	1
595349	33705	1
63101377	3572412	1
6688150613	378641967	1

Table XI D=336
x	y	z
55	3	1
6049	330	1
665335	36297	1
73180801	3992340	1
8049222775	439121103	1

This concludes Part VIII. Go to Part IX. Go back to Part VII.

Go back to homepage.

The Pellian Equation x2 −Dy2 = 1 from a Paired Sequence P(n) (Part VIII)

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

Tables of D and Pell (x,3,1) Triples

The Pellian Equation x² −Dy² = 1 from a Paired Sequence P(n) (Part VIII)