Number theory is rampant with easily stated conjectures that remain elusive for hundreds of years. A conjecture of this flavor was formulated comparatively recently, based on a circle of ideas of several mathematicians (Stothers, Mason, Szpiro, Masser, Oesterlé):

Suppose A and B are any positive integers with no common factor, and let C = A + B. Let N be the product of the prime factors of ABC. Then C < N².

This is, more or less, the ABC conjecture.

For instance, if A = 5 and B = 27, then C = 32 and N = 30. We have C < N².

Look at it another way: find an equation A + B = C with A and B having no common factors and log(C)/log(N) as large as possible. When A = 5 and B = 27, log(C)/log(N) &asymp 1.02. Try to find an example where, say, log(C)/log(N) > 1.6. Have fun!

Because of its links to certain other topics, this simple-sounding conjecture quickly became a central problem in modern number theory. If there is time, for instance, I'll describe how the ABC conjecture is related to the problem of finding integer solutions to certain equations, such as in Fermat's Last Theorem.

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