On X -coordinates of Pell equations which are repdigits
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RESEARCH
On X-coordinates of Pell equations which are repdigits Carlos A. Gómez1* , Florian Luca2,3,4,5 and Faith Shadow Zottor6 * Correspondence:
[email protected] Departamento de Matemáticas, Universidad del Valle 25360, Calle 13 # 100-00 Cali, Colombia Full list of author information is available at the end of the article 1
Abstract In this paper, we give an algorithm which finds, for an integer base b ≥ 2, all squarefree integers d ≥ 2 such that sequence of X -components {Xn }n≥1 of the Pell equation X 2 − dY 2 = ±1 has two members which are base b-repdigits. We implement this algorithm and find all the solutions to this problem for all bases b ∈ [2, 100]. Keywords: Repdigits, Linear forms in logarithms, Baker’s method Mathematics Subject Classification: 11B39, 11J86
1 Introduction For a positive integer base b ≥ 2 a repdigit is a positive integer N whose base brepresentation has a unique repeating digit. Letting a ∈ {1, . . . , b − 1} be the value of the repeating digit and m be the number of digits of N we have m b −1 . N =a b−1 When b = 10, we omit the base and simply call N a repdigit. Recently there has been a flurry of activity regarding finding all members of some classical sequence (Fibonacci numbers [11] and its generalisations [4], perfect powers [5], etc.) which are repdigits for some particular base b. Two classical sequences are the sequences of X- or Y -coordinates of the Pell equation X 2 − dY 2 = ±1 associated to a squarefree integer d ≥ 2. The first paper where the presence of X-coordinates of Pell equations was studied that we are aware of is [7]. There it is shown that if {Xn }n≥1 is the sequence of X-coordinates of the Pell equation X 2 − dY 2 = 1, then Xn is a repdigit for at most one value of n except for d = 2, 3, each of which has two solutions n for which Xn is a repdigit. In his MathSciNet review MR3491748 of [7], J. A. Vandehey writes: “The techniques (used in [7]) appear to be very specific to the base-10 case so that generalising to other bases would require a separate work of equal magnitude.” The problem was revisited in [9] where the general base b was treated. There it was shown that if Xn is a base b-repdigit for two values of n then 5
d ≤ exp((10b)10 ).
(1)
While the result has the theoretical merit of showing that the problem has a finite answer (even in effective form), the bound (1) is useless for practical computations. A close look
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at the proof from [9] shows that the bound (1) appeared when studying the equation m b −1 for a ∈ {1, . . . , b − 1} Xn = a b−1 2 − 1, the above equation with n even implies with n even. Since Xn = 2Xn/2
a(bm − 1) . b−1 Writing m = 3m0 + r for some r ∈ {0, 1, 2}, and putting (x, y) := (Xn/2 , bm0 ), the above equation becomes 2 −1= 2Xn/2
a(br y3 − 1) . (2) b−1 When a = b − 1, the above equation represents an elliptic curve and the integer points (x, y) on it can be effectively boun
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