The Pellian Equation x2 −Dy2 = 1 Revisited (Part I)
A Method for Generating Pellian Triples (x,y,1)
The Pellian equation is the Diophantine equation
x2 − Dy2 = z2 where z equals 1. The negative Pellian equation x2 − Dy2 = −1 is a topic of
Part II which follows a similar method as described in this page.
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√D)n ∕ 2√D)]
In addition, the method of converting a quadratic surd √D into continued fractions (pages 261-262) is also shown and may be also found online. The method is now available as JS computer code where the calculations are now in the hands of "the computer". However, a simpler non computer based method to generate the subsequent x and y values, starting with the known least solutions, is now the subject of this web page. The method involves the use of the following first mathematical expression which was later subsequently found to be equal to the second math expression:
RD = (n1 + √D)2 ∕n2 = x + y√D)
where n1 and n2 are arbitrary integers and
n1 is mostly less than the √D. RD is an irrational number or surd generated for each value of D and used for generating all subsequent values x and y. Initially the known least value solutions x and y are multiplied and rounded off by RD to generate new x and y solutions which are then tested to make sure they fit the Pell equation where z is equal to 1. If not the variables n1 and n2 are simply tweaked until the desired results are obtained. The RD that is obtained is then repeatedly multiplied by each row of x and y to generate the next row which is then tested in the Pell equation to confirm that z is equal to 1.
Relation of a Sequence Sn and the Pell Equation
At this point we can mention that the RDs in the following math expression:
RD = (n1 + √D)2
are related to a sequence Sn A002522 in the OEIS database and where the variable n1 are consecutive integers 1,2,3,4,5... and the variable Ds are the numbers having the formula:
D = n2 + 1
Examples where D equals 2, 5, 10 and 17 are shown in Tables II, IV, VII and XIII below.
It will be shown in Part III thru Part X that many of the surds may be derived from the OEIS or from new sequences (see homepage for a listing).
Tables of D and Pell (x,y,1) Triples
- Table I shows the triples for the Pell equation x2 − 2y2 = 1 using the first least solution (3,2,1) and R2 = (1 + √2)2 = 5.828427125 = 3 + 2√2.
- Table II shows the triples for the Pell equation x2 − 3y2 = 1 using the first least solution (2,1,1) and R3 = (1 + √3)2 ∕ 2 = 3.732050808 = 2 + √3 .
- Table III shows the triples for the Pell equation x2 − 5y2 = 1 using the first least solution (9,4,1) and R5 = (2 + √5)2 = 17.94427191 = 9 + 4√5.
- Note that only non square D values are allowed.
Table I D=2
| x |
y | z |
| 3 | 2 | 1 |
| 17 | 12 | 1 |
| 99 | 70 | 1 |
| 577 | 408 | 1 |
| 3363 | 2378 | 1 |
| 19601 | 13860 | 1 |
| 114243 | 80782 | 1 |
| 665850 | 470832 | 1 |
|
|
Table II D=3
| x |
y | z |
| 2 | 1 | 1 |
| 7 | 4 | 1 |
| 26 | 15 | 1 |
| 97 | 56 | 1 |
| 362 | 209 | 1 |
| 1351 | 780 | 1 |
| 5042 | 2911 | 1 |
| 18817 | 10864 | 1 |
|
|
Table III D=5
| x |
y | z |
| 9 | 4 | 1 |
| 161 | 72 | 1 |
| 2889 | 1292 | 1 |
| 51841 | 23184 | 1 |
| 930249 | 416020 | 1 |
| 16692641 | 7465176 | 1 |
| 299537289 | 133957148 | 1 |
| 5374978561 | 2403763488 | 1 |
|
- Table IV shows first eight triples for the Pell equation x2 − 6y2 = 1 using the first least solution (5,2,1) and R6 = (2 + √6)2 ∕ 2 = 9.898979486 = 5 + 2√6.
- Table V shows the triples for the Pell equation x2 − 7y2 = 1 using the first least solution (8,3,1) and R7 = (3 + √7)2 ∕ 2 = 15.93725393 = 8 + 3√7.
- Table VI shows the triples for the Pell equation x2 − 8y2 = 1 using the first least solution (3,1,1) and R8 = (2 + √8)2 ∕ 4 = 5.828427125 = 3 + √8.
Table IV D=6
| x |
y | z |
| 5 | 2 | 1 |
| 49 | 20 | 1 |
| 485 | 198 | 1 |
| 4801 | 1960 | 1 |
| 47525 | 19402 | 1 |
| 470449 | 192060 | 1 |
| 4656965 | 1901198 | 1 |
|
|
Table V D=7
| x |
y | z |
| 8 | 3 | 1 |
| 127 | 48 | 1 |
| 2024 | 765 | 1 |
| 32257 | 12192 | 1 |
| 514088 | 194307 | 1 |
| 8193151 | 3096720 | 1 |
| 130576328 | 49353213 | 1 |
|
|
Table VI D=8
| x |
y | z |
| 3 | 1 | 1 |
| 17 | 6 | 1 |
| 99 | 35 | 1 |
| 577 | 204 | 1 |
| 3363 | 1189 | 1 |
| 19601 | 6930 | 1 |
| 114243 | 40391 | 1 |
|
- Table VII shows first six triples for the Pell equation x2 − 10y2 = 1 using the first least solution (19,6,1) and R10 = (3 + √10)2 = 37.97366596 = 19 + 6√10.
- Table VIII shows the triples for the Pell equation x2 − 11y2 = 1 using the first least solution (10,6,1) and R11 = (3 + √11)2 ∕ 2 = 19.94987437 = 10 + 3√11.
- Table IX shows the triples for the Pell equation x2 − 12y2 = 1 using the first least solution (7,2,1) and R12 = (3 + √12)2 ∕ 3 = 13.92820323 = 7 + 2√12.
Table VII D=10
| x |
y | z |
| 19 | 6 | 1 |
| 721 | 228 | 1 |
| 27379 | 8658 | 1 |
| 1039681 | 328776 | 1 |
| 39480499 | 12484830 | 1 |
| 1499219281 | 474094764 | 1 |
|
|
Table VIII D=11
| x |
y | z |
| 10 | 3 | 1 |
| 199 | 60 | 1 |
| 3970 | 1197 | 1 |
| 79201 | 23880 | 1 |
| 1580050 | 476403 | 1 |
| 31521799 | 9504180 | 1 |
|
|
Table IX D=12
| x |
y | z |
| 7 | 2 | 1 |
| 97 | 28 | 1 |
| 1351 | 390 | 1 |
| 18817 | 5432 | 1 |
| 262987 | 75658 | 1 |
| 3650401 | 1053780 | 1 |
|
- Table X shows first six triples for the Pell equation x2 − 13y2 = 1 using the first least solution (649,180,1) and R13 = (3 + √13)2 ∕ 4 = 10.90832691 = (649 + 180√13)1 ∕ 3 . This is the first instance of a third root of an expression. Two others in the subset of D numbers were also found. See Part III.
- Table XI shows the triples for the Pell equation x2 − 14y2 = 1 using the first least solution (15,4,1) and R14 = (4 + √14)2 ∕ 2 = 29.96662955 = 15 + 4√14.
- Table XII shows the triples for the Pell equation x2 − 15y2 = 1 using the first least solution (4,1,1) and R15 = (3 + √15)2 ∕ 6 = 7.872983346 = 4 + √15.
- For D = 13 rows 2,3,5,6 are non Pell triples since x and y have non integer values.
Table XI D=13
| x |
y | z |
| 649 | 180 | 1 |
| 7079.5 | 1963.5 | 1 |
| 77225.5 | 21418.5 | 1 |
| 842401 | 233640 | 1 |
| 9189185.5 | 2548621.5 | 1 |
| 100238639.5 | 27801196.5 | 1 |
| 1093435849 | 303264540 | 1 |
|
|
Table XI D=14
| x |
y | z |
| 15 | 4 | 1 |
| 449 | 120 | 1 |
| 13455 | 3596 | 1 |
| 403201 | 107760 | 1 |
| 12082575 | 3229204 | 1 |
| 362074049 | 96768360 | 1 |
| 10850138895 | 2899821596 | 1 |
|
|
Table XII D=15
| x |
y | z |
| 4 | 1 | 1 |
| 31 | 8 | 1 |
| 244 | 63 | 1 |
| 1921 | 496 | 1 |
| 15124 | 3905 | 1 |
| 119071 | 30744 | 1 |
| 937444 | 242047 | 1 |
|
- Table XI can be revised by using (R13)3 =10.908326913 = 1297.9992295835 instead of R13 and multipling (R13)3, first by the least and then all subsequent solutions, a procedure which eliminates all non Pell solutions. The equation generated from this value of √D leads to a separate discussion and will be treated separately in Part III.
- Table XIII shows the first five triples for the Pell equation x2 − 17y2 = 1 using the first least solution (33,8,1) and R17 = (4 + √17)2 = 65.984845 = 33 + 8√17.
- Table XII shows the triples for the Pell equation x2 − 18y2 = 1 using the first least solution (17,4,1) and R18 = (4 + √18)2 ∕ 2 = 33.97056275 = 17 + 4√18.
Table XI D=13 (revised)
| x |
y | z |
| 649 | 180 | 1 |
| 842401 | 233640 | 1 |
| 1093435849 | 303264540 | 1 |
| 1419278889601 | 393637139280 | 1 |
|
|
Table XIII D=17
| x |
y | z |
| 33 | 8 | 1 |
| 2177 | 528 | 1 |
| 143649 | 34840 | 1 |
| 9478657 | 2298912 | 1 |
| 625447713 | 151693352 | 1 |
|
|
Table XIV D=18
| x |
y | z |
| 17 | 4 | 1 |
| 577 | 136 | 1 |
| 19601 | 4620 | 1 |
| 665857 | 156944 | 1 |
| 22619537 | 5331476 | 1 |
|
This concludes Part I. Go to Part II.
Go back to homepage.
Copyright © 2020 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com