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

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,6,1). x equal to 19 was described in Part I. The triple (19,6,1) belongs to the D value of 10 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 new sequence which can be generated using the equation for a pair of numbers P(n) = n(9n − 1), n(9n + 1) corresponding to the various Ds:

0, 0, 8, 10, 34, 38, 78, 84, 140, 148, 220, 230, 318, 330, 434, 448, 568, 584, 720, 738, 890, 910, ...

where the initial numbers of the sequence are zeros, i.e., 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 = (n₁ + √D)² ∕ ⅓n₁ = x + 6√D)

where:

n₁ is an even integers from 2, 4, 6, 8, ...
the values of adjacent xs in a pair is (18n₁ − 1, 18n₁ + 1).
D are the values from the above sequence starting at 8.

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 every D is an even number.

At this point it must be mentioned that the sequence shown above starts at the integer 3. The importance of this is that tables of least solution have D = 8 with only x=3 and y=1. However, here we have the second instance, of a D whose x and y having multiple values, in this case x=17 and y=6 and so this D is being included as the first entry below. *Note that the first instance is covered in Part VI.

Tables of D and Pell (x,6,1) Triples

Note: all the (n₁ + √D)² ∕ ½n₁ are present in pairs.
Table I shows the triples for the Pell equation x² − 8y² = 1 using R₈ = (3 + √8)² = 17 + 6√8 = 33.97056275
Table II shows the triples for the Pell equation x² − 10y² = 1 using R₁₀ = (3 + √10)² = 19 + 6√10 = 37.97366596

Table I D=8
x	y	z
17	6	1
577	204	1
19601	6930	1
665857	235416	1
22619537	7997214	1

Table II D=10
x	y	z
19	6	1
721	228	1
27379	8658	1
1039681	328776	1
39480499	12484830	1

Table III shows the triples for the Pell equation x² − 34y² = 1 using R₃₄ = (6 + √34)² ∕ 2 = 35 + 4√34 = 69.98571137
Table IV shows the triples for the Pell equation x² − 38y² = 1 using R₃₈ = (6 + √38)² ∕ 2 = 37 + 4√38 = 73.98648402

Table III D=34
x	y	z
35	6	1
2449	420	1
171395	29394	1
11995201	2057160	1
849492675	143971806	1

Table IV D=38
x	y	z
37	6	1
2737	444	1
202501	32850	1
14982337	2430456	1
1108490437	179820894	1

Table V shows the triples for the Pell equation x² − 78y² = 1 using R₇₈ = (9 + √78)² ∕ 3 = 53 + 6√78 = 105.9905652
Table VI shows the triples for the Pell equation x² − 84y² = 1 using R₈₄ = (9 + √84)² ∕ 3 = 55 + 6√84 = 109.9909083
Table VII shows the triples for the Pell equation x² − 140y² = 1 using R₁₄₀ = (12 + √140)² ∕ 4 = 71 + 6√140 = 141.9929574

Table V D=78
x	y	z
53	6	1
5617	636	1
595349	67410	1
63101377	7144824	1
6688150613	757283934	1

Table VI D=84
x	y	z
55	6	1
6049	660	1
665335	72594	1
73180801	7984680	1
8049222775	878242206	1

Table VII D=140
x	y	z
71	6	1
10081	852	1
14314311	120978	1
203253121	17178024	1
28860511751	2439158430	1

Table VIII shows the triples for the Pell equation x² − 148y² = 1 using R₁₄₈ = (12 + √148)² ∕ 4 = 73 + 6√148 = 145.9931504
Table IX shows the triples for the Pell equation x² − 220y² = 1 using R₂₂₀ = (15 + √220)² ∕ 5 = 89 + 6√220 = 177.9943818
Table X shows the triples for the Pell equation x² − 230y² = 1 using R₂₃₀ = (15 + √230)² ∕ 5 = 91 + 6√230 = 181.9945053

Table VIII D=148
x	y	z
73	6	1
10657	876	1
1555849	127890	1
227143297	18671064	1
33161365513	2725847454	1

Table IX D=220
x	y	z
89	6	1
15841	1068	1
2819609	190098	1
501874561	33836376	1
89330852249	6022684830	1

Table X D=230
x	y	z
91	6	1
16561	1092	1
3014011	198738	1
548533441	36169224	1
99830072251	6582600030	1

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

Go back to homepage.

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

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

Tables of D and Pell (x,6,1) Triples

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