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

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,10,1). x with the values 49, 51 and 99 are listed in the above articles with D values, 24, 26 and 98, respectively. 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 (not in the OEIS database) generated using the equation for a pair of numbers P(n) = n(25n − 1), n(25n + 1) corresponding to the various Ds:

0, 0, 24, 26, 98, 102, 222, 228, 396, 404, 620, 630, 894, 906, 1218, 1232, 1592, 1608, ...

where each pair uses the same value of n.

Accordingly, we can place each of these various D numbers in Table D along with their respective x values. All y values are 10.

Table D
n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
D	24	26	98	102	222	228	396	404	620	630	894	906	1218	1232
x	49	51	99	101	149	151	199	201	249	251	299	301	349	351

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

R_D = (5n₁ + √D)² ∕ n₁ = x + 10√D)

where:

n₁ are consecutive integers, 1, 2, 3, 4, 5, ...
the values of adjacent xs in a pair is (50n₁ − 1, 50n₁ + 1).
D are the values from the above sequence starting at 24.

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 twelve Ds of the above sequence and the accompanying triple tables generated for each even D.

At this point it must be mentioned that the sequence shown above starts at the integer 24. The importance of this is that tables of least solution have D = 24 with only x=5 and y=1. However, here we have another instance, of a D whose x and y having multiple values, in this case x=49 and y=10 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,10,1) Triples

Note: all the (5n₁ + √D)² ∕ n₁ are present in pairs.
Table I shows the triples for the Pell equation x² − 24y² = 1 using R₂₄ = (5 + √24)² = 49 + 10√24 = 97.98979486
Table II shows the triples for the Pell equation x² − 26y² = 1 using R₂₆ = (5 + √26)² = 51 + 10√26 = 101.9901951
Table III shows the triples for the Pell equation x² − 98y² = 1 using R₉₈ = (10 + √98)² ∕ 2 = 99 + 10√98 = 197.9949494

Table I D=24
x	y	z
49	10	1
4801	980	1
470449	96030	1
46099201	9409960	1

Table II D=26
x	y	z
51	10	1
5201	1020	1
530451	104030	1
54100801	10610040	1

Table III D=98
x	y	z
99	10	1
19601	1980	1
3880899	392030	1
768398401	77619960	1

Table IV shows the triples for the Pell equation x² − 102y² = 1 using R₁₀₂ = (10 + √102)² ∕ 2 = 101 + 10√102 = 201.9950494
Table V shows the triples for the Pell equation x² − 222y² = 1 using R₂₂₂ = (15 + √222)² ∕ 3 = 149 + 10√222 = 297.9966443
Table VI shows the triples for the Pell equation x² − 228y² = 1 using R₂₂₈ = (15 + √228)² ∕ 3 = 151 + 10√228 = 301.9966887

Table IV D=102
x	y	z
101	10	1
20401	2020	1
4120901	408030	1
832401601	82420040	1

Table V D=222
x	y	z
149	10	1
44401	2980	1
13231349	888030	1
3942897601	264629960	1

Table VI D=228
x	y	z
151	10	1
45601	3020	1
13771351	912030	1
4158902401	275430040	1

Table VII shows the triples for the Pell equation x² − 396y² = 1 using R₃₉₆ = (20 + √396)² ∕ 4 = 199 + 10√396 = 397.9974874
Table VIII shows the triples for the Pell equation x² − 404y² = 1 using R₄₀₄ = (20 + √404)² ∕ 4 = 201 + 10√404 = 401.9975124
Table IX shows the triples for the Pell equation x² − 620y² = 1 using R₆₂₀ = (25 + √620)² ∕ 5 = 249 + 10√620 = 497.997992

Table VII D=396
x	y	z
199	10	1
79201	3980	1
31521799	1584030	1
12545596801	630439960	1

Table VIII D=404
x	y	z
201	10	1
808011	4020	1
32481801	16166030	1
13057603201	649640040	1

Table IX D=620
x	y	z
249	10	1
124001	4980	1
61752249	2480030	1
30752496001	1235049960	1

Table X shows the triples for the Pell equation x² − 630y² = 1 using R₆₃₀ = (25 + √630)² ∕ 5 = 251 + 10√630 = 501.998008
Table XI shows the triples for the Pell equation x² − 894y² = 1 using R₈₉₄ = (30 + √894)² ∕ 6 = 299 + 10√894 = 597.9983278
Table XII shows the triples for the Pell equation x² − 906y² = 1 using R₉₀₆ = (30 + √906)² ∕ 6 = 301 + 10√906 = 601.9983389

Table X D=630
x	y	z
251	10	1
126001	5020	1
63252251	2520030	1
31752504001	1265050040	1

Table XI D=894
x	y	z
299	10	1
178801	5980	1
106922699	3576030	1
63939595201	2138459960	1

Table XI D=906
x	y	z
301	10	1
181201	6020	1
109082701	3624030	1
65667604801	2181660040	1

This concludes Part XI. Go to Part XIIA. Go back to Part X.

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

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

Tables of D and Pell (x,10,1) Triples

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