Really strong Carmichael numbers
Nombres de Carmichael réellement forts
Résumé
A composite number $n$ is called a Carmichael number if $a^{n-1}\equiv 1\pmod{n}$ for any integer $a$ coprime with $n$. D. H. Lehmer considered the class of these numbers $n$ such that $a^{(n-1)/2}\equiv \left(\frac{a}{n}\right)\pmod{n}$ for any integer $a$ coprime with $n$. Here $\left(\frac{a}{n}\right)$ denotes the Jacobi symbol. It turns out and it is shown by Lehmer himself that this class is empty. Here, we replace $\equiv\left(\frac{a}{n}\right)\pmod n$ in Lehmer's congruence by $\equiv 1\pmod n$ and get a new class which is not empty.
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