Even perfect numbers in generalized Pell sequences
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Lithuanian Mathematical Journal
Even perfect numbers in generalized Pell sequences Jhon J. Bravo1 and Jose L. Herrera Departamento de Matemáticas, Universidad del Cauca, Calle 5 No 4–70, Popayán, Colombia (e-mail: [email protected]; [email protected]) Received October 19, 2019; revised March 10, 2020
Abstract. In this paper, by using linear forms in logarithms and the Baker–Davenport reduction procedure we prove that there are no even perfect numbers appearing in generalized Pell sequences. We also deduce some interesting results involving generalized Pell numbers, which we believe are of independent interest. This paper continues a previous work that searched for perfect numbers in the classical Pell sequence. MSC: 11B39, 11J86 Keywords: generalized Pell number, perfect number, linear form in logarithms, reduction method
1 Introduction A perfect number is defined as any positive integer where the sum of its proper divisors equals the number. The first integers satisfying this property are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, . . .
(sequence A000396).
Euclid, over two thousand years ago, showed that 2p−1 (2p − 1) is an even perfect number whenever 2p − 1 is a prime. The prime numbers of the form 2p − 1 are known as Mersenne primes and require p itself to be prime. A much less obvious result, due to Euler, showed that all even perfect numbers are of the form 2p−1 (2p − 1) for some p such that 2p − 1 is a Mersenne prime. This is known as the Euclid–Euler theorem. Theorem [Euclid–Euler]. An even positive integer is a perfect number if and only if it has the form 2p−1 × (2p − 1) with 2p − 1 a prime. This theorem describes the relationship between perfect numbers and Mersenne primes. It is conjectured that there are infinitely many Mersenne primes. This is equivalent, by the Euclid–Euler theorem, to the conjecture that there are infinitely many even perfect numbers. It is also unknown whether there exists even a single odd perfect number. There are a lot of integer sequences used in number theory. For instance, the Fibonacci sequence (Fn )∞ n=0 is one of the most famous and curious numerical sequences in mathematics and has been widely studied in the literature. Also, there is the Pell sequence, which is as important as the Fibonacci sequence. The Pell sequence 1
The author was supported in part by Projects VRI ID 4689 (Universidad del Cauca) and Colciencias 110371250560.
c 2020 Springer Science+Business Media, LLC 0363-1672/20/6004-0001
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J.J. Bravo and J.L. Herrera
(Pn )∞ n=0 is defined by the recurrence Pn = 2Pn−1 + Pn−2 for n 2 with P0 = 0 and P1 = 1 as initial conditions. For the beauty and rich applications of these numbers and their relatives, we refer the reader to Koshy’s book [8]. Let φ(n) and σ(n) be the Euler function and the sum of divisors of the positive integer n, respectively. Note that a number n is perfect if σ(n) = 2n. There are many papers in the literature dealing with Diophantine equations involving arithmetic functions in members of a binary recur
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