Quote:
Originally posted by andi314
Can i then say that the tested number is prime
...
i thought that carmichael numbers are really rare and can be excluded by doing a little trail factoring.
...
but these numbers are eliminated by trial factoring up to 7000!!!???

There is a big mathematical difference between a statement that can be proved to be true and a statement that is true 99.9999...% of the time
but sometimes, even if very rarely, is false..
If one claims, or wants to say, simply that "[a certain number] is prime", that implies that it can be proven, absolutely and without any possible doubt, that the number is 100% definitely a prime number.
If, instead, one is referring to a number which has been proven by trial factoring to have no factors smaller than some limit (e.g., 7000 or even 7 trillion) and, in addition, passes the Fermat pseudoprime test for 4 (or even 4000) different prime bases, then one can correctly say that the number is a
probable prime, but one
cannot correctly say that the number is a "prime"
with no qualifying adjective ahead of the word "prime". So it's always necessary to refer to the latter example as a "probable prime" or "pseudoprime".