| Abstract:Let N be the set of all positive integers. Given n∈N, let φ(n) and σ{n) be the Euler function of n and the sum of divisors of n, respectively. We call n a Nicol number if n|φ(n)+σ(n), and a t-Nicol number if φ(n)+σ(n)=tn. Since C. A. Nicol formulated the famous Nicol conjecture in1966, Da-Zheng Lin and Ming-Zhi Zhang, F. Luca and J. Sandor, K. Harris had studied the Nicol numbers which have two, three, four different prime divisors. In this dissertation, we studied the Nicol numbers which have five different prime divisors.The main results of this thesis are divided into two parts. In the first part we show that the Nicol numbers which have five different prime divisors must be3-Nicol numbers or4-Nicol numbers. Moreover, we show that all but finitely many3-Nicol numbers which have five different prime divisors must be n=2α1·3·p3α3·p4α4·p5α5or n=2α15α2P3α3P4α4P5α5,p3≤59or n=2α17α2P3α3P4α4P5α5,P3≤19, where αi≥1, Pi are primes.In the second part we show that all but finitely many4-Nicol numbers which have five different prime divisors must be n=2α1·3·5·α3·7α4·qαs, where q>7is prime, and are natural numbers. |
