The Pellian Equation x² −Dy² = ±1 Revisited (Part II)

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

The Pellian equation x² − Dy² = 1 was covered in Part I. The negative Pellian equation x² − Dy² = −1 is a topic of this page which follows a similar method as described previously with some modifications. The least solutions of the negative Pell equation, however, are not posted in either Wikipedia (which has a small section describing this topic) or listed in Recreations in the Theory of Numbers by Albert H. Beiler (1966) as were their positive Pell solutions, but the following equations on page 253 may be used for their computation:

x = [(p + q√D)^2n-1 + (p − q√D)^2n-1 ∕ 2]
y = [(p + q√D)^2n-1 + (p − q√D)^2n-1 ∕ 2√D)]

In addition, the method of converting a quadratic surd √D into continued fractions (pages 261-262) are also methods that can be used to generate these least solutions. However, a simpler method to generate the subsequent x and y values, starting with the known least solutions for the positive Pell equation, is now the subject of this web page. The method involves the use of the following mathematical expressions:

R_Du = (n₁ + √D) ∕n₂
R_D = (n₁ + √D)² ∕n₂

where n₁ was computed and then found to be the same number as the least x of the negative least solution, viz., the x value in red in each of the odd numbered tables (a totally unexpected surprise) and n₂ is 1. R_Du is the non squared irrational number or surd obtained from R_D which was used to calculate the x and y in the regular Pell equation of Part I. Initially the known least value solutions x and y of the positive Pell equation is multiplied and rounded off by R_Du to generate a new x and y solutions with z = −1. This new row of values is then tested to make sure it fits the negative Pell equation.

Multiplication of this new row by R_Du produces a third row of positive Pell values and further repetition produces −1 values followed by +1 values. Once the desired number of rows is obtained we can backtrack and generate the approximate least solution values of the negative Pell equation by taking the initial +1 least solutions x and y and dividing by R_Du. It is simple and generates the least solution of the negative Pell equation which we can tabulate in a new table. This can be done by either extracting the x and y from the R_Du table, or starting anew using the newly obtained x and y values then multiplying thru repeatedly by R_D just as was done in Part I for the positive Pell equation.

Only the first few numbers D in Part I for which x² − Dy² = −1 is solvable are 2,5,10,13,17,26,29... the numbers in the sequence A031396 in the OEIS. The values obtained for the R_Ds from Part I all have n₂ = 1 except for the D = 13 having an n₂ of 4 and whose R_D is calculated in a slightly different manner. All the others with n₂ > 1 are not solvable using the method on this page thus making it consistent with known methods.

Tables of D and Pell (x,y,±1) and (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 R_2u = (1 + √2) = 2.414213562.
First row in red is calculated value of least solution.
Table II shows the triples for the Pell equation x² − 2y² = −1 using the first least solution (1,1,-1) and R₂ = (1 + √2)² = 5.828427125.

Table I D=2
x	y	z
1	1	-1
3	2	1
7	5	-1
17	12	1
41	29	-1
99	70	1
239	169	-1
577	408	1
1393	985	-1
3363	2378	1
8119	5741	-1

Table II D=2
x	y	z
1	1	-1
7	5	-1
41	29	-1
239	169	-1
1393	985	-1
8119	5741	-1
47321	33461	-1
275807	195025	-1

Table III shows first triples for the Pell equation x² − 5y² = ±1 using the first least solution (9,4,1) and R_5u = (2 + √5) = 4.236067977.
First row in red is calculated value of least solution.
Table IV shows the triples for the Pell equation x² − 5y² = −1 using the first least solution (2,1,-1) and R₅ = (2 + √5)² = 17.94427191.

Table III D=5
x	y	z
2	1	-1
9	4	1
38	17	-1
161	72	1
682	1305	-1
2889	1292	1
12238	5473	-1
51841	23184	1
219602	98209	-1
930249	416020	1
3940598	1762289	-1

Table IV D=5
x	y	z
2	1	-1
38	17	-1
682	305	-1
12238	5473	-1
219602	98209	-1
3940598	1762289	-1
70711162	31622993	-1
1268860318	567451585	-1

Table V shows first triples for the Pell equation x² − 10y² = ±1 using the first least solution (19,6,1) and R_10u = (3 + √10) = 6.16227766.
First row in red is calculated value of least solution.
Table VI shows the triples for the Pell equation x² − 10y² = −1 using the first least solution (3,1,1) and R₁₀ = (3 + √10)² = 37.97366596.

Table V D=10
x	y	z
3	1	-1
19	6	1
117	37	-1
721	228	1
4443	1405	-1
27379	8658	1
168717	53353	-1
1039681	328776	1
6406803	2026009	-1
39480499	12484830	1
243289797	76934989	-1

Table VI D=10
x	y	z
3	1	-1
19	6	-1
117	37	-1
721	228	-1
4443	1405	-1
27379	8658	-1
168717	53353	-1
1039681	328776	-1

Table I shows first triples for the Pell equation x² − 13y² = ±1 using the first least solution (649,180,1) and (R_13u)^{3∕ 2} = [(3 + √13)²) ∕ 4]^{3∕ 2} = 36.02775638.
Table II shows the triples for the Pell equation x² − 13y² = −1 using the first least solution (18,5,-1) and (R₁₃)³ = [(3 + √13)² ∕ 4]³ = 1297.9992295835.

Table I D=13
x	y	z
18	5	-1
649	180	1
23382	6485	-1
842401	233640	1
30349818	8417525	-1
1093435849	303264540	1
39394040382	10925940965	-1
1419278889601	393637139280	1

Table II D=13
x	y	z
18	5	-1
23382	6485	-1
30349818	8417525	-1
39394040382	10925940965	-1
51133434066018	14181862955045	-1

Table VII shows the first triples for the Pell equation x² − 17y² = ±1 using the first least solution (33,8,1) and R_17u = (4 + √17) = 8.123105626.
Table VIII shows the triples for the Pell equation x² − 17y² = −1 using the first least solution (4,1,-1) and R₁₇ = (4 + √17)² = 65.984845.

Table VII D=17
x	y	z
4	1	-1
33	8	1
268	65	-1
2177	528	1
17684	4289	-1
143649	34840	1
1166876	283009	-1
9478657	2298912	1

Table VIII D=17
x	y	z
4	1	-1
268	65	-1
17684	4289	-1
1166876	283009	-1
76996132	18674305	-1
5080577836	1232221121	-1

Table IX shows the first triples for the Pell equation x² − 26y² = ±1 using the first least solution (51,10,1) and R_26u = (5 + √26) = 10.09901951.
Table X shows the triples for the Pell equation x² − 26y² = −1 using the first least solution (5,1,-1) and R₂₆ = (5 + √26)² = 101.9901951.

Table IX D=26
x	y	z
5	1	-1
51	10	1
515	101	-1
5201	1020	1
52525	10301	-1
530451	104030	1
5357035	1050601	-1
54100801	10610040	1

Table X D=26
x	y	z
5	1	-1
515	101	-1
52525	10301	-1
5357035	1050601	-1
546365045	107151001	-1

This concludes Part II. Go back to Part I. Go to Part III.

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The Pellian Equation x2 −Dy2 = ±1 Revisited (Part II)

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

Tables of D and Pell (x,y,±1) and (x,y,−1) Triples

The Pellian Equation x² −Dy² = ±1 Revisited (Part II)