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.