In 1969, D. J. Lewis wrote a paper about Diophantine equations, in
which he wrote that the equation !!x^3+5=117y^3!! is known to have at most
18 solutions, but the exact number is not known. Two other
mathematicians studied the equation and, in 1971, they published a
short but difficult proof that there are no solutions. Finally, in
1973, another mathematician published a surprisingly short proof
that there are no solutions. The proof is:
The quantity !!x^3+5!! is
never a multiple of 9, but the quantity !!117y^3!! is always a multiple
of 9, so there are no solutions