On factorials in Perrin and Padovan sequences

Assume that wn is the nth term of either Padovan or Perrin sequence. In this paper, we solve the equation wn = m! completely.


Introduction
A number of mathematicians have been interested in Diophantine equations including both factorials and elements of linear recurrences such as Fibonacci, Tribonacci, and balancing numbers, etc. For example, Luca [6] proved that F n is a product of factorials only when n = 1, 2, 3, 6, 12 , where F n is the n th Fibonacci number.
Grossman and Luca [3] showed that the equation F n = m 1 ! + m 2 ! + · · · + m k ! has finitely many positive integers n for fixed k. In the same paper the solutions were determined for k ≤ 2 .
Marques and Lengyel [8] searched the factorials in Tribonacci sequence. They characterized the 2-adic order of Tribonacci numbers and then solved the equation completely. This was the first paper to find factorials in third-order linear recurrences. In this paper, we present the 2-adic order of Padovan and Perrin numbers. Afterwards, we investigate factorials in Perrin and Padovan sequences.
Before going further, we give the definitions of Perrin and Padovan numbers.
By the recurrence relations of Perrin and Padovan sequences, negative indices of these numbers can be obtained easily. The following is the list of few Padovan and Perrin numbers.
Perrin numbers were studied by several authors in the beginning of the nineteenth century (for details, see [9] This paper is divided into two parts. In the first part we give several necessary lemmas and 2 -adic orders of Perrin and Padovan numbers. In the second part, we solve the Diophantine equations R n = m!, P n = m! completely.
Let w n be the n th term of Padovan or Perrin sequences.

Auxiliary results
Before proceeding further, some lemmas will be needed. The next lemma gives additional formulas for Perrin and Padovan numbers.

Lemma 2.1 Let n, m be positive integers. Then
and follow.
Proof These formulas can be found in [11] (Proposition 2.2.). 2 The following lemma gives a recurrence relation with arithmetic progression for Perrin and Padovan numbers.

Lemma 2.2
Let n, s , and r be positive integers with 0 ≤ s ≤ r − 1 . Then we have where α , β , β are the roots of the equation Proof The identity can be proven in a way similar to [4]. 2 Since the Binet formula of a Perrin number is then formula (2.3) can be written as Now we introduce the following matrix notations: By the recurrence relation of the Perrin and Padovan sequences, one can easily check that CT n = T n+1 and CB n = B n+1 . Then we obtain that These facts give that
From now on, we resume the induction on t . Therefore, we can assume that P 7·2 t +j ≡ P j (mod 2 t+2 ) holds for integers t and j . Our aim is to show P 7·2 t+1 +j ≡ P j (mod 2 t+3 ).
We follow induction on j again. Assume that j = 1 . We can write where a t,j are positive integers satisfying the recurrence of the sequence {P n }. Now define The second formula in (2.5) gives where T (P ) n is the matrix whose entries are w n = P n in the matrix T n .
By the definitions of the vectors T Therefore, the second formula in (2.5) yields that Then follows as claimed. Assume that P 7·2 t+1 +j ≡ P j (mod 2 t+3 ) holds for j ≡ 1 (mod 7). Let j = 7k + 1 for k ∈ Z.
The other cases for the integer j can be proven similarly. 2

Lemma 2.4 For the integers j and t ≥ 1, we have
, Proof It can be proven in a way similar to the proof of the previous lemma. Therefore, we do not give the details. 2 The p-adic order ν p (r) of r is the exponent of the highest power of a prime p , which divides r . We provide a complete description of the 2-adic order of Perrin and Padovan numbers.

2
The following Lemma is about the 2-adic order of a Padovan number.
follows as claimed.
We separate the case into two subcases.
This inequality gives that 3 ≤ m ≤ 56 and then n ≤ 3.61m log m 2 + 2 ≤ 675. By using similar arguments, we obtain that 3 ≤ m ≤ 16 and n ≤ 123 for the equation R n = m!. A simple routine written in Mathematica shows that there is no solution for the equations P n = m! and R n = m! for the given interval.