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Perfect powers from products of terms in Lucas sequences. (English) Zbl 1137.11011

A Lucas sequence \(\{U_n\}_{n\geq 0}\) has terms defined by \(U_n=(\alpha^n-\beta^n)/(\alpha-\beta)\), where \(\alpha,\beta\) are the roots of an equation \(x^2-rx-s=0\) with \(r,s\) integers satisfying \(\Delta=r^2+4s\neq 0\). Equivalently, \(U_0=0\), \(U_1=1\) and \(U_{n+2}=rU_{n+1}+sU_n\) for \(n\geq 0\). If \(r=s=1\), then the Lucas sequence is reduced to the Fibonacci sequence \(\{F_n\}_{n\geq 0}\).
Assume that we are given a non-degenerate Lucas sequence \(\{U_n\}_{n\geq 0}\) (i.e. we are given \(r\) and \(s\) and the corresponding \(\alpha/\beta\) is not a root of unity) and a set of primes \(T=\{l_1,\dots,l_t\}\). Assume further that there exist \(m\) positive integers \(n_1,\ldots,n_m\), a prime number \(p>m\), a positive integer \(y\) and non-negative integers \(x_1,\dots,x_t\) such that
\[ U_{n_1}\cdots U_{n_m}=\pm l_1^{x_1}\cdots l_t^{x_t}y^p. \tag{\(*\)} \]
Then, according to the main theorem of the paper, there exists an effectively computable constant \(c\), depending only on the sequence and the set \(T\), such that \(n_i<c\) for \(i=1,\ldots,m\).
The very important feature of the method of proof of the above result is the following: Denote by \(q\) the greatest prime divisor of \(n_1\cdots n_m\). Then, it is proved that, either \(q\) belongs to a finite explicitly computable set of primes depending on the Lucas sequence and the set \(T\), or \(U_q\) satisfies a relation
\[ U_q=\pm A^{(q-1)/2}z^p.\tag{\(**\)} \]
where \(A\) is also depending on the sequence and on \(T\) and is explicitly computable.
Until recently equations of type \((**)\) were unapproachable. However, the method applied in the very important paper by the first, third and fourth author which made possible the explicit computation of all perfect powers in the Fibonacci sequence (as well as in the Lucas sequence defined by the same recurrence relation, but with initial terms 2,1) [Ann. Math. (2) 163, 969–1018 (2006; Zbl 1113.11021)] and also in a previously published paper of the authors [Int. J. Number Theory 1, 309–332 (2005; Zbl 1114.11014)] gives a strategy that hopefully can lead to the successful solution of equations \((**)\), as the authors explain in the concluding section of the paper. The three highly non-trivial and interesting applications given in the previous sections seriously support their explanations, of course. Two applications to the Fibonacci sequences are given. Roughly speaking, with \(U_n=F_n\), the authors solve \((*)\) in case that \(T=\emptyset\) or \(T=\) set of the 100 first primes.
They also prove that \(F_mF_n=y^p\) with \(1\leq m\leq n\) and \(p\) prime (including the case \(p=2\)) is possible only in the following cases: \(m=n\); \(m,n=1,2\); \(n=6\), \(m=1,2,3\); \(n=12\), \(m=1,2\). The third application is to “unidigital numbers”, i.e. numbers \(dU_n\), where \(U_n=(10^n-1)/(10-1)\), \((n\geq 1)\) and \(1\leq d\leq 9\) (hence, these numbers have \(dd\ldots d\) as their decimal representation). It is proved that the only pairs of such numbers whose products are perfect powers are \(\{dU_n,dU_n\}\) for every \(d\) and every \(n\); \(\{U_n,4U_n\}\) and \(\{U_n,9U_n\}\), for every \(n\); \(\{1,8\}\), \(\{2,4\}\), \(\{3,9\}\), \(\{4,8\}\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D61 Exponential Diophantine equations
11A63 Radix representation; digital problems
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