A Study on Arithmetic Functions and Diophantine Equations Associated with Balancing and Related Sequences

Abstract

A Lucas sequence is a binary recurrence sequences that includes as special cases, the Pell, the associatedPell, the balancing, the Lucasbalancing, the balancinglike and the associated balancinglike sequences. Arithmetic functions of some of these sequences result in interesting inequalities. In particular, σk(Bn) ≤ Bσk(n), where σk denotes the sum of the kth power of divisors function and Bn is the nth balancing number. Repdigits are positive integers with one distinct digit, that is, a single digit appears in the unit place, decimal place and so on. The Euler functions of Pell numbers has no repdigit with at least two digits. However, it is not known whether the Euler function of any associated Pell number is a repdigit with more than one digit. Moreover, if the Euler function of an associated Pell number is a repdigit, then its index must be odd, and the repdigit consists of the digits 4 or 8; in case it contains the digit 4, then its index is an odd prime or its square. The third term of every balancinglike sequence is one less than a square, while, by choice, the second term may or may not be one less than a square and no other term of any balancinglike sequence has this property. Furthermore, except the first and second terms, no other term of any balancinglike sequence is one more than a square. There are just 2 perfect powers in sums and differences of two balancing numbers and in any balancinglike sequence, the number of perfect squares in the sums and differences of two balancinglike numbers is always finite. BrocardRamanujan identity consists of finding positive integer solutions of the equation n! + 1 = m2 and the only known solutions are (n,m) = (4, 5), (5, 11), (7, 71). This problem can be generalized by replacing the positive integers in n! and m by balancinglike or associated balancinglike numbers. The revised problem has sometimes no solution and sometimes finitely many solutions.