The Pellian Equation x2 −Dy2 = 1 from a Paired Sequence P(n) (Part IX)
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,8,1). x equal to 33 was described in Part I. In addition, the triple (33,8,1) belongs to D value of 17 in the articles listed 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
A157716 which can be generated using the equation for a pair of numbers P(n) = n(16n − 1), n(16n + 1) corresponding to the various Ds:
0, 0, 15, 17, 62, 66, 141, 147, 252, 260, 395, 405, 570, 582, 777, 791, 1016, 1032, 1287, 1305, 1590, 1610, ...
where an extra zero has been added to the sequence so that 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:
RD = (4n1 + √D)2 ∕ n1 = x + 8√D)
where:
n1 are consecutive integers, 1, 2, 3, 4, 5, ...
the values of adjacent xs in a pair is (32n1 − 1, 32n1 + 1).
D are the values from the above sequence starting at 15.
then multiplying and rounding off each row of triples generated by the RD 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 D where a D may be even number or odd.
At this point it must be mentioned that the sequence shown above starts at the integer 15. The importance of this is that tables of least solution have D = 15 with only x=4 and y=1. However, here we have the third instance, of a D whose x and y having multiple values, in this case x=31 and y=8 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,8,1) Triples
- Note: all the (n1 + √D)2 ∕ ½n1 are present in pairs.
- Table I shows the triples for the Pell equation x2 − 15y2 = 1 using R15 = (4 + √15)2 = 31 + 8√15 = 61.98386677
- Table II shows the triples for the Pell equation x2 − 17y2 = 1 using R17 = (4 + √17)2 = 33 + 8√17 = 65.984845005
- Table III shows the triples for the Pell equation x2 − 62y2 = 1 using R62 = (8 + √62)2 ∕ 2 = 63 + 8√62 = 125.992062992
Table I D=15
| x |
y | z |
| 31 | 8 | 1 |
| 1921 | 496 | 1 |
| 119071 | 30744 | 1 |
| 7380481 | 1905632 | 1 |
|
|
Table II D=17
| x |
y | z |
| 33 | 8 | 1 |
| 2177 | 528 | 1 |
| 143649 | 34840 | 1 |
| 9478657 | 2298912 | 1 |
|
|
Table III D=62
| x |
y | z |
| 63 | 8 | 1 |
| 6937 | 1008 | 1 |
| 999999 | 127000 | 1 |
| 125991937 | 16000992 | 1 |
|
- Table IV shows the triples for the Pell equation x2 − 66y2 = 1 using R66 = (8 + √66)2 ∕ 2 = 65 + 8√66 = 129.9923072
- Table V shows the triples for the Pell equation x2 − 141y2 = 1 using R141 = (12 + √141)2 ∕ 3 = 95 + 8√141 = 189.9947367
- Table VI shows the triples for the Pell equation x2 − 147y2 = 1 using R147 = (12 + √147)2 ∕ 3 = 97 + 8√147 = 193.9948452
Table IV D=66
| x |
y | z |
| 65 | 8 | 1 |
| 8449 | 1040 | 1 |
| 1098305 | 135192 | 1 |
| 1142771201 | 17573920 | 1 |
|
|
Table V D=141
| x |
y | z |
| 95 | 8 | 1 |
| 18049 | 1520 | 1 |
| 3429215 | 288792 | 1 |
| 651532801 | 54868960 | 1 |
|
|
Table VI D=147
| x |
y | z |
| 97 | 8 | 1 |
| 18817 | 1552 | 1 |
| 3650401 | 301080 | 1 |
| 708158977 | 58407968 | 1 |
|
- Table VII shows the triples for the Pell equation x2 − 252y2 = 1 using R252 = (16 + √252)2 ∕ 4 = 127 + 8√252 = 253.9960629
- Table VIII shows the triples for the Pell equation x2 − 260y2 = 1 using R260 = (16 + √260)2 ∕ 4 = 129 + 8√260 = 257.996124
- Table IX shows the triples for the Pell equation x2 − 395y2 = 1 using R395 = (20 + √395)2 ∕ 5 = 159 + 8√395 = 317.9968553
Table VII D=252
| x |
y | z |
| 127 | 8 | 1 |
| 32257 | 2032 | 1 |
| 8193151 | 516120 | 1 |
| 2081028097 | 131092448 | 1 |
|
|
Table VIII D=260
| x |
y | z |
| 129 | 8 | 1 |
| 33281 | 2064 | 1 |
| 8586369 | 532504 | 1 |
| 2215249921 | 137383968 | 1 |
|
|
Table IX D=395
| x |
y | z |
| 159 | 8 | 1 |
| 50561 | 2544 | 1 |
| 16078239 | 808984 | 1 |
| 51128289441 | 257254368 | 1 |
|
- Table X shows the triples for the Pell equation x2 − 405y2 = 1 using R405 = (20 + √405)2 ∕ 5 = 161 + 8√405 = 321.9968944
- Table XI shows the triples for the Pell equation x2 − 570y2 = 1 using R582 = (24 + √570)2 ∕ 6 = 191 + 8√570 = 381.9973822
- Table XII shows the triples for the Pell equation x2 − 582y2 = 1 using R582 = (24 + √582)2 ∕ 6 = 193 + 8√582 = 385.9974093
Table X D=405
| x |
y | z |
| 161 | 8 | 1 |
| 51841 | 2576 | 1 |
| 16692641 | 829464 | 1 |
| 5374978561 | 267084832 | 1 |
|
|
Table XI D=570
| x |
y | z |
| 191 | 8 | 1 |
| 72961 | 3056 | 1 |
| 27870911 | 1167384 | 1 |
| 10646615041 | 445937632 | 1 |
|
|
Table XI D=582
| x |
y | z |
| 193 | 8 | 1 |
| 74497 | 3088 | 1 |
| 28755649 | 1191960 | 1 |
| 11099606017 | 460093472 | 1 |
|
This concludes Part IX. Go to Part X. Go back to Part VIII.
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Copyright © 2020 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com