Let {Un} n≥0 and {Vm} m≥0 be two linear recurrence sequences. We establish an asymptotic formula for the number of integers c in the range [−x, x] which can be represented as differences Un − Vm. In particular, the density of such integers is 0.
In this paper, we completely solve the Diophantine equation Fn +Fm = 2 a 1 + 2 a 2 + 2 a 3 + 2 a 4 + 2 a 5 , where F k denotes the k-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport reduction method, we use p-adic versions of both tools.
Let $$\{U_n\}_{n \ge 0}$$ { U n } n ≥ 0 and $$\{V_m\}_{m \ge 0}$$ { V m } m ≥ 0 be two linear recurrence sequences. We establish an asymptotic formula for the number of integers c in the range $$[-x, x]$$ [ - x , x ] which can be represented as differences $$ U_n - V_m$$ U n - V m . In particular, the density of such integers is 0.
Let ( U n ) n ∈ N (U_n)_{n\in \mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B B and N 0 N_0 such that for any b , c ∈ Z b,c\in \mathbb {Z} with b > B b> B the equation U n − b m = c U_n - b^m = c has at most two distinct solutions ( n , m ) ∈ N 2 (n,m)\in \mathbb {N}^2 with n ≥ N 0 n\geq N_0 and m ≥ 1 m\geq 1 . Moreover, we apply our result to the special case of Tribonacci numbers given by T 1 = T 2 = 1 T_1= T_2=1 , T 3 = 2 T_3=2 and T n = T n − 1 + T n − 2 + T n − 3 T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n ≥ 4 n\geq 4 . By means of the LLL-algorithm and continued fraction reduction we are able to prove N 0 = 2 N_0=2 and B = e 438 B=e^{438} . The corresponding reduction algorithm is implemented in Sage.
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