A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with $Z_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ contains exactly two components. The primality of Fermat number is also discussed.

The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \mathbb{N}$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $min\{x,y,z\}>1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p/8]}\left(modp\right)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.