Sums of S-units in the solution sets of generalized Pell equations

In this paper, we give various finiteness results concerning solutions of generalized Pell equations representable as sums of S-units with a fixed number of terms. In case of one term, our result is effective, while in case of more terms, we are able to bound the number of solutions.


Introduction.
There are many papers about equations of the form where (U n ) ∞ n=0 is a linear recurrence sequence, and z 1 , . . . , z k are integers with prime factors coming from a fixed finite set of primes. Here we only refer to the recent papers Guzman-Sanchez and Luca [8], Bertók et al. [1], Bérczes et al. [2], and the (many) references there, where several and various finiteness results have been proved. We mention that there are also many results in the literature where other related problems are discussed. For example, Bravo et al. [4] considered a problem connected to sums of terms of a recurrence sequence yielding perfect powers (also see the references there).
In this paper, we consider the problem of representability of solutions of generalized Pell equations as a fixed term sum of integers with prime factors coming from some finite set of primes. As we shall see, this problem is closely related to Eq. (1). In fact, the problem is more general: it turns out that we need to find sums of the form z 1 + · · · + z k in unions of recurrence sequences, rather than in only one fixed sequence. We note that there are some closely related results in the literature. We mention only two recent papers Luca and 2. New results. Before formulating our theorem, we need to introduce some new notation.
The equation Further, for γ ∈ Q, write h(γ) for the maximum of the absolute values of the numerator and the denominator of γ. Finally, for a non-zero integer m, let ω(m) denote the number of distinct prime divisors of |m|. Now we can give our results about sums of S-units in the solution sets of generalized Pell equations. In the particular case of 'one-term' sums, our theorem is effective, that is, we are able to bound all the parameters involved. In the general case, we can bound only the number of solutions. Theorem 2.1. Use the above notation, and let k ≥ 1. Then there are at most for any 0 < j ≤ k and 1 ≤ i 1 < · · · < i j ≤ k, and would yield infinitely many solutions for the inclusion (4).

The proof of Theorem 2.1.
To prove our theorem, we need several lemmas. The first one describes the solutions of Eq. (2) in the particular, but very important case t = 1.
Then all positive integer solutions u, v of (5) are given by Proof. The statement is [12, Theorem 7.26, p. 354].
Before formulating our further lemmas, we need to introduce some notation concerning recurrence sequences. Let A, B be integers with B = 0, and let U 0 , U 1 be integers such that at least one of them is non-zero. Then the sequence U = (U n ) n≥0 satisfying the relation is called a binary linear recurrence sequence. We shall also use the notation U = U (A, B, U 0 , U 1 ) for the sequence. The characteristic polynomial of U is defined by Write α and β for the roots of f (x). The sequence U is called degenerate if α/β is a root of unity; otherwise it is called non-degenerate. It is well-known that if U is non-degenerate, then we have Our second lemma shows that the sets of the coordinates of the solutions of Eq. (2) are unions of finitely many non-degenerate binary linear recurrence sequences. We note that this assertion is long and well-known qualitatively. However, we do not know any source where this statement is explicitly formulated (let alone the paper of Liptai [10] which is in Hungarian). In fact, we shall only need the case concerning solutions with gcd(x, y) = 1. However, we find the general case of possible independent interest. For a non-negative integer m, write τ (m) for the number of divisors of |m|. with some binary recurrence sequences .
Here I and G where c 3 is an effectively computable constant depending only on τ (t), while c 4 is an effectively computable constant depending only on d and t. Further, for the solutions (x, y) of (2) with gcd(x, y) = 1, the same conclusion holds with I < c 5 and (8), where c 5 is an effectively computable constant depending only on ω(t).
Proof. Obviously, we may restrict to positive integer solutions of (2). So let (p, q) be a positive solution of (2). Then the norm N (p + √ dq) of the algebraic integer p + √ dq is t in the field Q( √ d). By [9, Lemma 5], we know that there are only finitely many pairwise non-associate algebraic integers and their number I can be bounded in terms of τ (t); further, under the assumption gcd(p, q) = 1, even in terms of ω(t). It is well-known (see, e.g., [14, Chapter A]) that we may assume here that where c 6 is an effectively computable constant depending only on d, t. Thus there exist algebraic integers U i + √ dV i with N (U i + √ dV i ) = t and max(|U i |, |V i |) < c 6 (i = 1, . . . , I) such that . We immediately get that N (ν) = 1. Thus Lemma 3.1 yields that For simplicity, we assume that ν = (u 0 + √ dv 0 ) m with some m ≥ 0 since all the other cases are similar (or can be excluded by our assumption that p and q are positive). Then we have Putting from these assertions, we obtain Hence, as α, β are roots of the polynomial x 2 − 2u 0 x + 1 (also in view of (7)), we get that p and q are elements of the recurrence sequences G = G(A, B, G Finally, note that it is obvious that the terms of these recurrence sequences are solutions of (2). Hence our claim follows.
We shall also need a recent finiteness result of Bérczes et al. [2] concerning the number of terms of recurrence sequences representable as k-term sums of S-units. Lemma 3.3. Let U n be a non-degenerate binary linear recurrence sequence as in (6), and suppose that the characteristic polynomial of U n has irrational roots. Then for any fixed k ≥ 1, Eq. (1) is solvable in z 1 , . . . , z k ∈ U S at most for finitely many n. Further, the number of indices n for which (1) is solvable for this fixed k, can be bounded by an effectively computable constant depending only on and k.
Proof. The statement is a simple consequence of [2, Theorem 1] and its proof. Note that the statement in [2] concerns only the case where z 1 , . . . , z k ∈ U S ∩Z, however, from the proof it is clear that this more general formulation is also valid.
Our last lemma is a deep result concerning the finiteness of the solutions of S-unit equations. For its formulation, we need to introduce some further notation.
Let K be an algebraic number field, and let S = {P 1 , . . . , P } be a finite set of prime ideals of K. Write U S for the S-units in K, that is, for the set of those α ∈ K for which the principal fractional ideal (α) can be represented as (α) = P b1 1 · · · P b (b 1 , . . . , b ∈ Z). By the (naive) height h(γ) of an element γ ∈ K we mean the maximum of the absolute values of the coefficients of the defining primitive polynomial of γ in Z[x]. Note that for γ ∈ Q, h(γ) is just the maximum of the absolute values of the numerator and denominator of γ.