The Pellian Equation x² −Dy² = 1 A General Method (Part XIIA)

A Method for Generating Pellian Triples from Paired Sequences P(n) (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:

The tables in these two articles show a series of numbers of which I will focus on those triples of the type (x,y,1) starting at (3,2,1) and D = 2. It will be shown here that we can continue generating all those values of x not listed in these articles by employing a general sequence method where the specific seqeunces from this general method may or may not be in the OEIS database. The sequences uses the expression consisting of a pair of numbers and the corresponding x values while keeping y constant:

corresponding to the various D and x values in the sequence where each pair uses the same value of n, and a is equal to the square of y. Thus, this method is a way of obtaining the Pellian triples from the set of square numbers S = {(y₁)²,(y₂)²,(y₃)²,...} of which a miniscule number are found in the tables of the above articles.

However, as a check once we find D and its corresponding x value we can backtrack and find y according to the following:

where y is the floor(y) value, i.e., rounded down to the nearest integer equal to the square root of a.

Accordingly, Tables D and E shows the calculation of P(n) and x(n) for a, viz., the square numbers 1,4,9,16,25,36,49,64,81 and n 1 to 5, producing the following tabulated results for D and x. Values of y from the floor(y) are calculated from the previous two rows, another way of verifying for the value of y. Moreover, when y = 1 the D and xs are repeated so that we can eliminate the n(n − 2) and n − 1 and just focus on the n(n + 2) and n + 1 . The sequence for these D numbers (where y = 1 in Table D) is listed in the OEIS database entry A005563 as n(n + 2).

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

where:

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 twelve Ds from table D and the accompanying three row triple tables generated for each D.

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

• Note: The Pell equation for each example is x² − 2y² = 1 plus all (n₁y + √D)² ∕ 2n₁ occur in pairs.
• Table I shows the triples for R₂ = (2 + √2)² ∕ 2 = 3 + 2√2 = 5.828427125
• Table II shows the triples for R₆ = (2 + √6)² ∕ 2 = 5 + 2√6 = 9.898979486
• Table III shows the triples for R₁₂ = (4 + √12)² ∕ 4 = 7 + 2√12 = 13.92820323

• Table IV shows the triples for R₂₀ = (4 + √20)² ∕ 4 = 9 + 2√20 = 17.94427191
• Table V shows the triples for R₃₀ = (6 + √30)² ∕ 6 = 11 + 2√30 = 21.95445115
• Table VI shows the triples for R₄₂ = (6 + √42)² ∕ 6 = 13 + 2√42 = 25.9614814

• Note: The Pell equation for each example is x² − 3y² = 1.
• Table VII shows the triples for R₇ = (3 + √7)² ∕ 2 = 8 + 3√7 = 15.93725393
• Table VIII shows the triples for R₁₁ = (3 + √11)² ∕ 2 = 10 + 3√11 = 19.94987437
• Table IX shows the triples for R₃₂ = (6 + √32)² ∕ 4 = 17 + 3√32 = 33.97056275

• Table X shows the triples for R₄₀ = (6 + √40)² ∕ 4 = 19 + 3√40 = 37.97366596
• Table XI shows the triples for R₇₅ = (9 + √75)² ∕ 6 = 26 + 3√75 = 51.98076211
• Table XII shows the triples for the R₈₇ = (9 + √87)² ∕ 6 = 28 + 3√87 = 55.98213716

This concludes Part XII. Go to Part XIIb. Go back to Part XI.

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