On members of Lucas sequences which are products of factorials
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On members of Lucas sequences which are products of factorials Shanta Laishram1
· Florian Luca2,3,4 · Mark Sias5
Received: 24 September 2019 / Accepted: 25 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We show that if {Un }n≥0 is a Lucas sequence, then the largest n such that |Un | = m 1 !m 2 ! · · · m k ! with 1 ≤ m 1 ≤ m 2 ≤ · · · ≤ m k satisfies n < 62,000. When the roots of the Lucas sequence are real, we have n ∈ {1, 2, 3, 4, 6, 12}. As a consequence, we show that if {X n }n≥1 is the sequence of X -coordinates of a Pell equation X 2 − dY 2 = ±1 with a non-zero integer d > 1, then X n = m! implies n = 1. Keywords Lucas sequence · Factorials · Pell equations · Primes in arithmetic progressions · Primitive divisors · abc conjecture Mathematics Subject Classification Primary 11B39 · 11B65; Secondary 11D72 · 11D45
Communicated by Adrian Constantin.
B
Shanta Laishram [email protected] Florian Luca [email protected] Mark Sias [email protected]
1
Stat-Math Unit, Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi 110016, India
2
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
3
Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia
4
Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico
5
Department of Pure and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa
123
S. Laishram et al.
1 Introduction Let r , s be coprime nonzero integers with r 2 + 4s = 0. Let α, β be the roots of the quadratic equation x 2 − r x − s = 0 and assume without loss of generality that |α| ≥ β|. We assume further that α/β is not a root of 1. The Lucas sequences {Un }n≥0 and {Vn }n≥0 of parameters (r , s) are given by Un =
αn − β n α−β
and
Vn = α n + β n
for all
n ≥ 0.
Alternatively, they can be defined recursively by U0 = 0, U1 = 1, V0 = 2 and V1 = r and both recurrences Un+2 = rUn+1 + sUn and Vn+2 = r Vn+1 + sVn
hold for all n ≥ 0.
Let P F :=
⎧ ⎨ ⎩
±
k
m j ! : k ≥ 1and1 ≤ m 1 ≤ m 2 ≤ · · · ≤ m k
j=1
⎫ ⎬ ⎭
be the set of integers which are the product of factorials. Members of P F are sometimes called Jordan-Polya numbers and several recent papers investigate their arithmetic properties. For example, the counting function of P F (that is, the cardinality of the set P F ∩ [1, x] for positive real numbers x) was recently studied in [4], while the occurrence of perfect powers in P F was studied in [2]. In [6], it was shown that if t ≥ 1 is any fixed integer, then the Diophantine equation t
Un i ∈ P F
(1)
i=1
has only finitely many positive integer solutions 1 ≤ n 1 ≤ n 2 ≤ · · · ≤ n t and they are all effectively computable. When (r , s) = (1, 1) then Un = Fn is the nth Fibonacci number. For this particular case, it was shown in [7] that the largest solution of Eq. (1) in which t is also an indeterminate but with the condition on the indices 1 ≤ n 1 < n 2 < · · · < n t is F1 F2 F3 F4 F5 F6 F8 F10 F12 = 11! Similar
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