
Glossary: Prime Pages: Top 5000: 
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 F_{n}. 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 summingIn 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 F_{n} < A/(2^{n} log 2)over the nonnegative 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
Chris K. Caldwell © 19992015 (all rights reserved)
