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.
Let \( \{F_n\}_{n\geq 0} \) be the sequence of Fibonacci numbers and let \(p\) be a prime. For an integer \(c\) we write \(m_{F,p}(c)\) for the number of distinct representations of \(c\) as \(F_k-p^\ell\) with \(k\ge 2\) and \(\ell\ge 0\). We prove that \(m_{F,p}(c)\le 4\).
Let $$ (P_n)_{n\ge 0}$$
(
P
n
)
n
≥
0
be the sequence of Perrin numbers defined by ternary relation $$ P_0=3 $$
P
0
=
3
, $$ P_1=0 $$
P
1
=
0
, $$ P_2=2 $$
P
2
=
2
, and $$ P_{n+3}=P_{n+1}+P_n $$
P
n
+
3
=
P
n
+
1
+
P
n
for all $$ n\ge 0 $$
n
≥
0
. In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and the reduction procedure involving the theory of continued fractions, to explicitly determine all Perrin numbers that are concatenations of two repeated digit numbers.
Let (P n ) n≥0 be the sequence of Perrin numbers defined by ternary relation P 0 = 3, P 1 = 0, P 2 = 2, and P n+3 = P n+1 + P n for all n ≥ 0. In this paper, we use Baker's theory for nonzero linear forms in logarithms of algebraic numbers and the reduction procedure involving the theory of continued fractions, to explicitly determine all Perrin numbers that are concatenations of two distinct repeated digit numbers.
Let (P n ) n≥0 be the sequence of Perrin numbers defined by ternary relation P 0 = 3, P 1 = 0, P 2 = 2, and P n+3 = P n+1 + P n for all n ≥ 0. In this paper, we use Baker's theory for nonzero linear forms in logarithms of algebraic numbers and the reduction procedure involving the theory of continued fractions, to explicitly determine all Perrin numbers that are concatenations of two distinct repeated digit numbers.
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