In 1980, C. Pomerance, J. Selfridge, and S. S. Wagstaff defined a *pseudoprime to the base a* to be any composite odd $n$ such that $n \mid a^{n-1} - 1$.


More recently, in 2013, S. S. Wagstaff referred to such numbers as ``Fermat pseudoprimes.''


Are either of the following known to be true?---


(1) Every Lucas sequence of the form $a^{n} - 1$ contains at least one pseudoprime to the base $a$.


(2) There exists a value of $a$ for which no odd composite $n$ divides $a^{n-1} - 1$.


If neither are definitively true, could someone point me to any current research on item (2)


Thank you.



On pseudoprimes to the base $a$ (Fermat pseudoprimes)
(1) is true.


Let $p$ be a prime such that $p\nmid (a-1)a$ and $\frac{a^p-1}{a-1}$ is composite. Then $\frac{a^p-1}{a-1}$ is a base-$a$ pseudoprime.


Also, if $q$ is a Carmichael number comprime to $(a-1)a$, then both $q$ and $\frac{a^q-1}{a-1}$ are base-$a$ pseudoprimes.


