TITLE: Vertical shift of polynomials with integer roots
QUESTION [9 upvotes]: Given a polynomial with integer roots, is it possible to add an integer to the polynomial so the roots of the new polynomial are also integers.
Apparently it is possible for some polynomials.
For example $$ x^2-6x+5$$ and $$x^2-6x+8$$ satisfy our conditions.
Third order polynomials such as $$x^3-12x^2+41x-30$$ and $$x^3-12x^2+41x-42$$
satisfy the conditions as well.
My question is what is the highest  possible degree of polynomials satisfying the said conditions.

REPLY [6 votes]: The roots of such polynomial pairs are essentially the ideal solutions of the Prouhet–Tarry–Escott problem . 
According to above wiki entry,
ideals solutions are known for $3 \le n \le 10$ and for $n = 12$.
No ideal solution is known for $n = 11$ or for $n \ge 13$.
Following solution for $n = 12$ is extracted from
Chen Shuwen's
 Equal sums of like powers page. Let 
$$\begin{align}
(a)_{i=1}^{12} &= (1,12,25,66,91,130,174,213,238,279,292,303)\\
(b)_{i=1}^{12} &= (4,6,31,58,105,117,187,199,246,273,298,300)
\end{align}
$$
and consider following two polynomials of degree $12$,
$$A(x) = \prod_{i=1}^{12} (x - a_i)\quad\text{ and }\quad
B(x) = \prod_{i=1}^{12} (x - b_i)$$
We have
$$B(x) - A(x) = 67440294559676054016000$$