Carmichael showed for sufficiently large $L$, that $F_L$ has at least one prime divisor that is $pm 1({rm mod}, L)$. For a given $F_L$, we will show that a product of distinct odd prime divisors with that congruence condition is a Fibonacci pseudoprime. Such pseudoprimes can be used in an attempt, here unsuccessful, to find an example of a Baillie-PSW pseudoprime, i.e. an odd Fibonacci pseudoprime that is congruent to $pm 2({rm mod}, 5)$ and is also a base-2 pseudoprime.
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