Proving a conjecture can be extremely difficult, whereas a single counterexample is enough to refute it. But first you have to find one! With the conjectures proposed by Fermat and Euler (see the article " The exception that does not prove the rule"), attempting direct calculations by hand soon brings you up against a wall. A computer—or even a supercomputer—may be essential.
Computers have played a decisive role in solving several major mathematical questions in recent decades. Yet not every program is efficient: some take too long or require enormous amounts of memory. Many problems also run into combinatorial explosion: as soon as a parameter grows even slightly too large, the number of cases to examine becomes colossal!
A general method? ----------------------
In recent years, reinforcement-learning methods in artificial intelligence have produced spectacular results. Computers that, in a sense, acted as their own teachers have even defeated the world's greatest Go players! So why not use the same principles to hunt for counterexamples, particularly in graph theory?
Here is a broad outline of how it works. The agent (the computer) generates a construction and receives a score for its performance. By repeating the process several times, it eventually "learns" how to achieve a better score in this "game." If its score exceeds that of the best conjectured construction, a counterexample has been found.