If polynomial $P(x)$ has integer coefficients and at least three integer roots, then $P(x)+5^m$ has no more than one integer root for $m\geq 1$

I've been doing some polynomial excersises lately and in that one I got completly stuck.

Let $$m \geqslant 1$$ be natural number and $$P(x)$$ polynomial with integer coefficients which has at least three different integer roots. Prove that $$P(x)+5^m$$ has no more than one integer root.

At first I considered the easiest case: $$(x-x_1)(x-x_2)(x-x_3)+5$$, but it did not turned out in anything helpful, so I am seeking for some clues on how to crack that problem.

Also, I'd like to ask for as elementary hint/solution as possible since this question is from (inactive) high school contest. https://om.mimuw.edu.pl/static/app_main/problems/om48_1.pdf

• I haven't followed up far enough to make this a hint, but I'd start with the fact that for a monic polynomialthe constant term is the product of the roots. Nov 19, 2020 at 20:11
• Sorry, I misunderstood the question. Nov 19, 2020 at 21:10
• Could be of interest: artofproblemsolving.com/community/c6h133921p757901
– Sil
Nov 19, 2020 at 21:51

Suppose $$r_1$$, $$r_2$$, $$r_3$$ are three distinct integer roots of $$P(x)$$, and $$y_1$$ and $$y_2$$ are integer roots of $$P(x) + 5^m$$. Thus $$P(x) = Q(x)(x-r_1)(x-r_2)(x-r_3)$$ where $$Q(x)$$ also has integer coefficients. Now $$P(y_i) = -5^m = Q(y_i) (y_i - r_1)(y_i-r_2)(y_i-r_3)$$, so $$y_i - r_j$$ are integers that divide $$5^m$$, in particular they are $$\pm$$ powers of $$5$$. And $$y_1 - y_2 = (y_1 - r_1)-(y_2-r_1) = (y_1-r_2)-(y_2-r_2) = (y_1-r_3)-(y_2-r_3)$$ is written as the difference between two of those in three different ways. But that is impossible unless $$y_1 - y_2 = 0$$.
• How do we know $Q(x)$ has integer coefficients? Nov 19, 2020 at 21:31
• Long division of integer polynomial by $x-r_i$ produces integer polynomial. Nov 19, 2020 at 21:56
Let's look at things backwards: Suppose $$Q(x)$$ has integer coefficients and two integer roots; we need to show that $$Q(x)-5^m$$ cannot have three integer roots.
By translation, we may assume that one of the roots of $$Q$$ is positive and the other is at $$x=0$$. Let's let the positive root be at $$x=n$$, so that $$Q(x)=x(x-n)R(x)$$. Now if $$Q(r)=5^m$$, then both $$|r|$$ and $$|r-n|$$ are powers of $$5$$, say $$r=5^a\sigma$$ and $$n-r=5^b\tau$$, with $$\sigma,\tau\in\{1,-1\}$$. This implies $$n=5^a\sigma+5^b\tau$$, with a $$+1$$ for the sign of the larger power (since $$n\gt0$$). But no positive number $$n$$ can be written as a sum or difference of two powers of $$5$$ in more than one way (most cannot be written as a sum or difference of two powers of $$5$$ at all), so there are at most two possibilities for $$r$$ (i.e., either the larger power or the smaller power).