A heuristic is something "providing aid in the direction of the solution of a problem but otherwise unjustified or incapable of justification." So heuristic arguments are used to show what we might later attempt to prove, or what we might expect to find in a computer run. They are, at best, educated guesses.
For example, suppose one wondered if there are infinitely many Fermat primes Fn. We might argue the answer should be no heuristically as follows.
By the prime number theorem we can guess that the
probability of a "random" number n being prime is at
most A/log n for some real number A. So summing
In the above naive heuristic argument several unjustified
assumptions were made. For example, we assumed that the Fermat numbers behave enough like "random numbers" to make the above argument. However, Fermat numbers have special divisibility properties (see fermat divisors).
A/log Fn < A/(2n
log 2) over the non-negative integers we see we should expect at most A/log 2 Fermat primes, a finite number!"
If we apply the same argument above to the Mersenne
numbers, then we get a divergent sum, so it seems likely
that there are infinitely many Mersenne primes.
See Also: Conjecture, OpenQuestion