How many times should Fermat's primality test be applied? For a prime number $ p $, and an $ a $ such that $\ \  1<a<p-1 $,
we have: $\ \ \ a^{p-1} \equiv 1\ \pmod p $
To test the primality of a number, this is applied multiple times for randomly chosen $ a $ until either the number of iterations are up or the above equivalence becomes untrue, meaning the given number is composite. If the number of iterations are up, then the number is flagged as possibly prime.
Clearly, an upper limit on the number of iterations is $ p - 3 $ as there are $ p - 3 $ unique values for $ a $ to take.
However, $ p-3 $ seems excessive and  all computer implementations or resources I've seen for this algorithm (I haven't seen too many) always leave it up to the 'user' to set the number of iterations. 
I was wondering, is there a formula or a guideline that gives the ideal number of iterations to be performed for Fermat's primality test?

The weak Fermat-pseudoprime-test is a good start, but there are infinite many composite numbers passing the test for every base coprime to the number, namely the Carmichael numbers.
However, if your number is large (say $100$ digits or more), and if it is a "random number" (no special form) and you choose a random base, the probability that the number is prime is already very high if the number passes the test.
The strong-Fermat-pseudoprime-test is much better. If a number passes the test with $N$ random bases, the probability that it is a composite number is at most $(\frac{1}{4})^N$. If the number fails the test, it must be composite.
If you want to be absolutely sure that a number is prime, either use the Adleman-Pomerance-Rumely-test (APR) or Elliptic-Curve-Primality-Proving. The APR-test is implemented in most programs with long-integer-arithmetic, for example in PARI/GP.
