Abstract
Suppose that f(x)=x(x−k), where k is an odd positive integer. First, an infinite digraph Gk=(V,E) is defined, where the vertex set is V=ℤ and the edge set is E={(x,y)∣x,y∈ℤ,f(x)=f(2y)}. Then the following results are proved: if k=1, then the digraph Gk is weakly connected; if p is a safe prime, i.e., both p and q=(p−1)∕2 are primes, then the number wp of weakly connected components of the digraph Gp is 2. Finally, a conjecture that there are infinitely many primes p such that wp=2 is presented.
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