## Abstract

Let *N* be a positive integer, *N* is called an *α-Korselt number* (equivalently *α* is said an *N-Korselt base*) if *α*
_{2}
*p* − *α*
_{1} divides *α*
_{2}
*N* − *α*
_{1} for every prime divisor *p* of *N.* By the *Korselt set* of *N* over *N* is an *α*-Korselt number.

In this paper we determine explicitly for a given prime number *q* and an integer *l* ∈ ℕ \{0, 1}, the set *q ^{l}* -Korselt bases in ℚ and others in ℤ. The case of

*l*= 2.

Moreover, we show that each nonzero rational *α* is an *N*-Korselt base for infinitely many numbers *N* = *q ^{l}* where

*q*is a prime number and

*l*∈ ℕ.