$a,b,c \in \mathbf{Z}$ such that $a^7+b^7+c^7=45$ Do there exist integers $a,b,c$ such that $a^7+b^7+c^7=45$?
[I have an ugly argument for a negative answer, is it possible to give a "manual" solution?]

The seventh powers modulo $49$ are $0,\pm 1,\pm 18,\pm 19.$ There is no way to combine three of these to get $45$ modulo $49$.
