I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? What would be conditions for such conjectures? For example, there shouldn't have been extensive search already, and it should be (widely?) believed to be false.
For instance, I think the Riemann hypothesis is a bad example (much investigated and not believed to be wrong), but Firoozbakht's conjecture might be a good canditate (believed to be wrong, and not so famous).
Are there conjectures which fulfill those, or other conditions of that kind?
Edit: A very good example has just been published, namely the Boolean Pythagorean triples problem, where the researchers have used a computer to check roughly one trillion different possible solutions and found a negative result (see more in the Nature News&Views article).
Edit2: Another very nice example has just been presented at Numberphile, asking whether or not numbers can be written as the sum of three cubed numbers.