4个不同素因子时的Nicol问题

The Nicol question with four different prime divisors

主要讨论了4个不同素因子的Nicol数,得到了如下结果:(1)4个不同素因子的Nicol数只可能是3-Nicol数或4-Nicol数,同时也证明了Nicol猜想在4个不同素因子情况下是正确的;(2)4个不同素因子的4-Nicol数只能是2α1.3α2.5α3.pα4,p≥7为素数;(3)4个不同素因子的3-Nicol数只能是下列形式之一:n=2α1.7α2.11α3.pα4或n=2α1.7α2.13α3.pα4或n=2α1.5α2.pα3.qα4,p≤41,p<q或n=2α1.3α2.pα3.qα4,p,q为素数且5≤p<q。

In this disertation,we investigate the Nicol numbers with four different prime divisors. The main results are as follows： （1）The Nieol numbers with four different prime divisors must be 3-Nicol numbers or 4- Nicol numbers. At the same time, we prove the Nicol＇s conjecture with four different prime divisors. （2）The 4- Nicol numbers with four different prime divisors must be of the form andis a prime. （3）The 3-Nicol numbers with four different prime divisors must be of the form： n=2^α1·7^α2·11^α3·P^α4orn=2^α1·7^α2·13^α3·P^α4orn=2^α1·5^α2·P^α3·q^α4,P≤41.P〈qorn=2^α1·3^α2·P^α3·q^α4,P.qand q are primes and 5 ≤ p 〈 q.