TITLE: How to find the value of positive integers $a$-through-$h$
QUESTION [5 upvotes]: If the equation $(x-a)(x-b)(x-c)(x-d)(x-e)(x-f)(x-g) = hx$ has seven positive integer roots, and $a,b,c,d,e,f,g,h$ are positive integers too, how can we find them?

REPLY [4 votes]: According to Shen Chuwen's Equal Products and Equal Sums of Like Powers, this is the only known solution as of the year 2001:
$$(x - 666) (x - 663) (x - 616) (x - 595) (x - 558) (x - 497) (x - 480) - 1327233600 x \\
 = (x - 672) (x - 651) (x - 630) (x - 578) (x - 568) (x - 495) (x - 481).$$
A strategy for finding these would be to look for partial Prouhet-Tarry-Escott solutions in 7 variables that work only up to $5$th power, which gives $\prod_{i=1}^7 (x-a_i) - \prod_{i=1}^7 (x-b_i) = Cx + D$ for some integer constants $C,D$ with $C \ne 0$.  Then apply the substitution $x = y-D/C$ to clear the constant coefficient $D$, rescale to clear denominators and possibly negative signs, and voilà!  
The main thing that can go wrong is if $D/C$ lies in the smallest interval containing all the $a_i$ and $b_i$, which causes the end result to contain non-positive integers.