The Pellian Equation x² −Dy² = 1 from the Sequence (n + 1)² − 1 (Part IV)

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√D)ⁿ ∕ 2√D)]

The tables in these two articles show a series of numbers of which I will focus on those triples of the type (x,1,1). x equal to 2, 3 and 4 were described in Part I and all three follow a general pattern. The triples (2,1,1), (3,1,1) and (4,1,1) belong to the three D values 3, 8 and 15, respectively, as shown in the two articles 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 A005563 which can be generated using the equation (n + 1)² − 1:

0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, ...

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

R_D = (n₁ + √D)² ∕2n₁ = x + √D)

where:

n₁ are consecutive integers, 1, 2, 3, 4,...
x = n₁ + 1
D are the values from the above sequence starting at 3.

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 nine Ds of the above sequence and the accompanying triple tables generated for each D where an even number D follows and odd number D.

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

Table I shows the triples for the Pell equation x² − 3y² = 1 using R₃ = (1 + √3)² ∕2 = 2 + √3 = 3.732050808
Table II shows the triples for the Pell equation x² − 8y² = 1 using R₈ = (2 + √8)² ∕4 = 3 + √8 = 5.828427125
Table III shows the triples for the Pell equation x² − 15y² = 1 using R₁₅ = (3 + √15)² ∕6 = 4 + √15 = 7.872983346

Table I D=3
x	y	z
2	1	1
7	4	1
26	15	1
97	56	1
362	209	1

Table II D=8
x	y	z
3	1	1
17	6	1
99	35	1
577	204	1
3363	1189	1

Table III D=15
x	y	z
4	1	1
31	8	1
244	63	1
1921	496	1
15124	3905	1

Table IV shows the triples for the Pell equation x² − 24y² = 1 using R₂₄ = (4 + √24)² ∕8 = 5 + √24 = 9.898979486
Table V shows the triples for the Pell equation x² − 35y² = 1 using R₃₅ = (5 + √35)² ∕10 = 6 + √35 = 11.91607978
Table VI shows the triples for the Pell equation x² − 48y² = 1 using R₄₈ = (6 + √48)² ∕12 = 7 + √48 = 13.92820323

Table IV D=24
x	y	z
5	1	1
49	10	1
485	99	1
4801	980	1
47525	9701	1

Table V D=35
x	y	z
6	1	1
71	12	1
846	143	1
10081	1704	1
120126	20305	1

Table VI D=48
x	y	z
7	1	1
97	14	1
1351	195	1
18817	2716	1
262087	37829	1

Table VII shows the triples for the Pell equation x² − 63y² = 1 using R₆₃ = (7 + √63)² ∕8 = 8 + √63 = 15.93725393
Table VIII shows the triples for the Pell equation x² − 80y² = 1 using R₈₀ = (8 + √80)² ∕10 = 9 + √80 = 17.94427191
Table IX shows the triples for the Pell equation x² − 99y² = 1 using R₉₉ = (9 + √99)² ∕12 = 10 + √99 = 19.94987437

Table VII D=63
x	y	z
8	1	1
49	10	1
485	99	1
4801	980	1
47525	9701	1

Table VIII D=80
x	y	z
9	1	1
161	18	1
2889	323	1
51841	5796	1
930249	104005	1

Table IX D=99
x	y	z
10	1	1
199	20	1
3970	399	1
79201	7960	1
1580050	158801	1

This concludes Part IV. Go to Part V. Go to Part I.

Go back to homepage.

The Pellian Equation x2 −Dy2 = 1 from the Sequence (n + 1)2 − 1 (Part IV)

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

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

The Pellian Equation x² −Dy² = 1 from the Sequence (n + 1)² − 1 (Part IV)