TITLE: Innocent-looking Diophantine equation with smallest solution of the order $10^{50}$?
QUESTION [5 upvotes]: Recently someone mentioned to me that there is a diophantine equation that looks very simple and innocent, but the smallest solution involves numbers of the order $10^{50}$ or something like this. The equation is probably in either 1,2, or 3 varaibles. It has low coefficients, probably all 1 or 2. And the degree is low also, probably 4 or less.
Is there such an equation?
Edit: I think the equation might have been studied by Fermat, but I'm not sure.

REPLY [3 votes]: Here's a monster. The smallest integer solution to,
$$(x + a)^7 + (x - a)^7 + (2x + b)^7 + (2x - b)^7 + \\(-x - c)^7 + (-x + 
        c)^7 + (-2x - d)^7 + (-2x + d)^7 \\= 14^7(a^6 + 2b^6 - c^6 - 2d^6)^7$$
has $x$ with $\color{red}{1179\; \text{digits}}$. The variables $a,b,c,d$ are,
$$292565171139318137956759657471297,\\ 863420822620431936290192229011966,\\ 
534407060429869176086407612538177,\\ 859793943610761912321826231621886$$
and 
$$\small{x =481563304865430516682423843723465575123177045754683810551700\dots \approx \color{red}{4.81 \times 10^{1178}}}$$
Ajai Choudhry found this using an elliptic curve, which may explain the large values.