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

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,4,1). x equal to 9, 15 and 17 were described in Part I and all three follow a general pattern. The triples (9,4,1), (15,4,1) and (17,4,1) belong to the three D values 5, 14 and 18, 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 A074378 which can be generated using the equation for a pair of numbers P(n) = n(4n − 1), n(4n + 1) corresponding to the various Ds:

0, 0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333...

where an extra zero has been added to the sequence so as to form the pair 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 ∕ ½n₁ = x + 4√D)

where:

n₁ are even integers, 2, 4, 6, 8, ...
the values of adjacent xs in a pair is (8n₁ − 1, 8n₁ + 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 ten Ds of the above sequence and the accompanying triple tables generated for each D where D may be even or odd.

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 = 3 with only x=2 and y=1. However, here we have the first instance of a D whose x and y having multiple values, in this case x=7 and y=4 and so this D is being included as the first entry below.

Tables of D and Pell (x,4,1) Triples

Note: all the (n₁ + √D)² ∕ ½n₁ are present in pairs.
Table I shows the triples for the Pell equation x² − 3y² = 1 using R₃ = (2 + √3)² = 7 + 4√3 = 13.92820323
Table II shows the triples for the Pell equation x² − 5y² = 1 using R₅ = (2 + √5)² = 9 + 4√5 = 17.94427191
Table III shows the triples for the Pell equation x² − 14y² = 1 using R₁₄ = (4 + √14)² ∕ 2 = 15 + 4√14 = 29.96662955
Table IV shows the triples for the Pell equation x² − 18y² = 1 using R₁₈ = (4 + √18)² ∕ 2 = 17 + 4√18 = 33.97056275

Table I D=3
x	y	z
7	4	1
97	56	1
1351	12780	1
18817	10864	1
262087	151316	1

Table II D=5
x	y	z
9	4	1
161	72	1
2889	1292	1
51841	23184	1
930249	416020	1

Table III D=14
x	y	z
15	4	1
449	120	1
13455	3596	1
403201	107760	1
12082575	3229204	1

Table IV D=18
x	y	z
17	4	1
577	136	1
19601	4620	1
665857	156944	1
22619537	5331476	1

Table V shows the triples for the Pell equation x² − 33y² = 1 using R₃₃ = (6 + √33)² ∕ 3 = 23 + 4√33 = 45.97825059
Table VI shows the triples for the Pell equation x² − 39y² = 1 using R₃₉ = (6 + √39)² ∕ 3 = 25 + 4√39 = 49.97999199
Table VII shows the triples for the Pell equation x² − 60y² = 1 using R₆₀ = (8 + √60)² ∕ 4 = 31 + 4√60 = 61.98386677

Table V D=33
x	y	z
23	4	1
1057	184	1
48599	8460	1
2234497	388976	1
102738263	17884436	1

Table VI D=39
x	y	z
25	4	1
1249	200	1
62425	9996	1
3120001	499600	1
155937625	24970004	1

Table VII D=60
x	y	z
31	4	1
1921	248	1
119071	15372	1
7380481	952816	1
457470751	59059220	1

Table VIII shows the triples for the Pell equation x² − 68y² = 1 using R₆₈ = (8 + √68)² ∕ 4 = 33 + 4√68 = 65.984845
Table IX shows the triples for the Pell equation x² − 95y² = 1 using R₉₅ = (10 + √95)² ∕ 5 = 39 + 4√95 = 77.98717738
Table X shows the triples for the Pell equation x² − 105y² = 1 using R₁₀₅ = (10 + √105)² ∕ 5 = 41 + 4√105 = 81.98780306

Table VIII D=68
x	y	z
33	4	1
2177	264	1
143649	17420	1
9478657	1149456	1
625447713	75846676	1

Table IX D=95
x	y	z
39	4	1
3041	312	1
237159	24332	1
18495361	1897584	1
1442400999	147987220	1

Table X D=105
x	y	z
41	4	1
3361	328	1
275561	26892	1
22592641	2204816	1
1852321001	180768020	1

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

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

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

Tables of D and Pell (x,4,1) Triples

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