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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

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  • $\begingroup$ 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. $\endgroup$ Nov 19, 2020 at 20:11
  • $\begingroup$ Sorry, I misunderstood the question. $\endgroup$ Nov 19, 2020 at 21:10
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    $\begingroup$ Could be of interest: artofproblemsolving.com/community/c6h133921p757901 $\endgroup$
    – Sil
    Nov 19, 2020 at 21:51

2 Answers 2

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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 $y_1 - y_2 = 0$.

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  • $\begingroup$ How do we know $Q(x)$ has integer coefficients? $\endgroup$ Nov 19, 2020 at 21:31
  • $\begingroup$ Long division of integer polynomial by $x-r_i$ produces integer polynomial. $\endgroup$ Nov 19, 2020 at 21:56
  • $\begingroup$ Could you explain a little more why the last part of your propf works? $\endgroup$
    – 1qwertyyyy
    Nov 19, 2020 at 22:55
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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).

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