For integers $a\ge b\ge 2$, is $f(a,b) = a^b + b^a$ injective?
Solution 1:
Integers of the form $a^b + b^a$, $a,b > 1$ were named Leyland numbers by Crandall & Pomerance in honor of Paul Leyland. See https://oeis.org/A076980. See also http://en.wikipedia.org/wiki/Leyland_number for more about these numbers, including a project to factor them.