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

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,2,1). x equal to 3, 5 and 7 were described in Part I and all three follow a general pattern. The triples (3,2,1), (5,2,1) and (7,2,1) belong to the three D values 2, 6 and 12, 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 A002378 which can be generated using the equation n(n + 1):

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210...

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

R_D = (n₁ + √D)² ∕n₁ = x + 2√D)

where:

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

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 every D is an even number.

Tables of D and Pell (x,2,1) Triples

Table I shows the triples for the Pell equation x² − 2y² = 1 using R₂ = (1 + √2)² = 3 + 2√2 = 5.828427125
Table II shows the triples for the Pell equation x² − 6y² = 1 using R₆ = (2 + √6)² ∕2 = 5 + 2√6 = 9.898979486
Table III shows the triples for the Pell equation x² − 12y² = 1 using R₁₂ = (3 + √12)² ∕3 = 7 + 2√12 = 13.92820323

Table I D=2
x	y	z
3	2	1
17	12	1
99	70	1
577	408	1
3363	2378	1

Table II D=6
x	y	z
5	2	1
49	20	1
485	198	1
4801	1960	1
47525	19402	1

Table III D=12
x	y	z
7	2	1
97	28	1
1351	390	1
18817	5432	1
262987	75658	1

Table IV shows the triples for the Pell equation x² − 20y² = 1 using R₂₀ = (4 + √20)² ∕4 = 9 + 2√20 = 17.94427191
Table V shows the triples for the Pell equation x² − 30y² = 1 using R₃₀ = (5 + √30)² ∕5 = 11 + 2√30 = 21.95445115
Table VI shows the triples for the Pell equation x² − 42y² = 1 using R₄₂ = (6 + √42)² ∕6 = 13 + 2√42 = 25.9614814

Table IV D=20
x	y	z
9	2	1
161	36	1
2889	646	1
51841	11592	1
930249	208010	1

Table V D=30
x	y	z
11	2	1
241	44	1
5291	966	1
116161	21208	1
2550251	465610	1

Table VI D=42
x	y	z
13	2	1
3373	52	1
8749	1350	1
227137	35048	1
5896813	909898	1

Table VII shows the triples for the Pell equation x² − 56y² = 1 using R₅₆ = (7 + √56)² ∕7 = 15 + 2√56 = 29.9662955
Table VIII shows the triples for the Pell equation x² − 72y² = 1 using R₇₂ = (8 + √72)² ∕8 = 17 + 2√72 = 33.97056275
Table IX shows the triples for the Pell equation x² − 90y² = 1 using R₉₀ = (9 + √90)² ∕9 = 19 + 2√90 = 37.97366596

Table VII D=56
x	y	z
15	2	1
449	60	1
13455	1798	1
403201	53880	1
12082575	1614602	1

Table VIII D=72
x	y	z
17	2	1
577	68	1
19601	2310	1
665857	78372	1
22619537	2665738	1

Table IX D=90
x	y	z
19	2	1
721	76	1
27379	2886	1
1039681	109592	1
39480499	4161610	1

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

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

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

Tables of D and Pell (x,2,1) Triples

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