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Let N be a positive integer, $${\mathbb {A}}$$ be a nonempty subset of $${\mathbb {Q}}$$ and $$\alpha =\dfrac{\alpha _{1}}{\alpha _{2}}\in {\mathbb {A}}{\setminus } \{0,N\}$$. $$\alpha $$ is called an N-Korselt base (equivalently N is said an $$\alpha $$-Korselt number) if $$\alpha _{2}p-\alpha _{1}$$ is a divisor of $$\alpha _{2}N-\alpha _{1}$$ for every prime p dividing N. The set of all Korselt bases of N in $${\mathbb {A}}$$ is called the $${\mathbb {A}}$$-Korselt set of N and is simply denoted by $${\mathbb {A}}$$-$$\mathcal {KS}(N)$$. Let p and q be two distinct prime numbers. In this paper, we study the $${\mathbb {Q}}$$-Korselt bases of pq, where we give in detail how to provide $${\mathbb {Q}}$$-$$\mathcal {KS}(pq)$$. Consequently, we finish the incomplete characterization of the Korselt set of pq over $${\mathbb {Z}}$$ given in Ghanmi (JAMA 42:2752–2762, 2018), by supplying the set $${\mathbb {Z}}$$-$$\mathcal {KS}(pq)$$ when $$q u003c2p$$.

The $${\mathbb {Q}}$$-Korselt set of $$\mathrm {pq}$$