Fermat's primality test for base 2 permits Poulet numbers to pass the test, as follows: $(2^x - 2)/x$. Fermat's primality test in different bases will act as a sieve for eliminating most pseudo primes from passing the test, unless the numbers are Carmichael numbers.
I ran an experiment for the following formula $(5^x - 3^x - 2^x)/x$ and it seems to eliminate all but Carmichael numbers, without having to check different bases.
I was capable of running the experiment until 10000 only (due to my lack of computing calculation power).
Does anyone know about this formula and whether it still holds forever?
The answer to this question was answered here (inspired by the comments above): https://mathoverflow.net/questions/369430/fermats-little-theorem-poulet-numbers-carmichael-numbers-and-primes Yes the formula is known and yes there are smaller numbers than 25326001.