Suppose $r$ is a rational number and for $k > 2$, consider $0\leqslant a_1< a_2<\cdots \leqslant a_k$. Also, for $n > 2$ and assume that we are not interesting the case of $n = 4 = k$, then there exists only finitely many solutions of $x$ in set of integers and $y$ in set of rational numbers to the equation $ r + (x-a_1)(x-a_2)\cdots(x-a_k) = y^n $ and all the solutions satisfy $\max\{H(x), H(y)\} < C$, where $C$, is an effectively computable constant depending only on $n$, $r$, and $a_i$'s. Here $r$ is an integer and not a perfect $n$-th power. Generalize the truth of this statement and show the solutions existence with $k$ bound.
$edit$: We recall that the height $H(α)$ of an algebraic number α is the maximum of the absolute values of the integer coefficients in its minimal defining polynomial In particular, if α is a rational integer, then $H(α) = |α|$ and if α is a rational number and not equal to $zero$ Then$ H(α) = max (|p|, |q|)$.
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