On the multiplicity in Pillai's problem with Fibonacci numbers and powers of a fixed prime

Abstract

Let {Fn} n≥0 be the sequence of Fibonacci numbers and let p be a prime. For an integer c we write mF,p(c) for the number of distinct representations of c as F k − p ℓ with k ≥ 2 and ℓ ≥ 0. We prove that mF,p(c) ≤ 4.

On the Diophantine equation $U_n-b^m = c$

“…As a special case of Theorem 1 we get the following result for the Tribonacci numbers, where the technical condition is checked directly in the proof (Section 6). Note that the case of Fibonacci numbers has recently been considered by Batte et al [2]. Remark 11.…”

“…Therefore, a and α are multiplicatively independent. Finally, we check that there are no unwanted solutions to (2). Assume that α z − 1 = a x α y with z ∈ N, x, y ∈ Q and −1 < x < 0.…”

“…Below we have enclosed the Sage code that was used to determine the small solutions to T n1 −T n2 = b m1 −b m2 in Section 6, as well as the code that was used to determine the bounds for the reduction rounds in Table 3. if b^x * (b^y -1) == diff: m2 = x m1 = x + y c = t2 -b^x print("T_{",n1,"} -T_{",n2,"} =",b,"^{",m2,"} (",b,"^{",y,"}-1), \\quad c&=", c, "= T_{",n1,"} -",b,"^{",m1,"} = T_{",n2,"} -",b,"^{",m2,"}; \\\\") # reduction steps alpha = n(solve(x^3 -x^2 -x -1 == 0, x, solution_dict=True) [2][x], digits=2000) a = 1/(-alpha^2 + 4*alpha -1)…”