Fermat Quartics and Serre's Challenge Curve

Fermat himself determined all rational points on the curve X^4 + Y^4 = 1, namely (X,Y) = (0,1),(0,-1),(1,0),(-1,0). The generalisation of this to X^n + Y^n = 1 has been well publicised! Instead, we consider a different generalisation, to all curves of the form: a X^4 + b Y^4 = c, where a,b,c are integers, and in particular the case: X^4 + Y^4 = c. For many values of c one can find all rational points using elementary congruence arguments or by maps to elliptic curves. However, there remain occasional values of c, such as c = 17, 82, ... etc, for which these elementary techniques are unsuccessful. Serre asks how one might in particular try to solve the case c = 17. We discuss various alternative lines of attack to try to deal with these exceptional values of c, including the line of attack which allowed Flynn and Wetherell finally to solve the case c = 17 in 1999.