Let $p(x)$ be a monic $2003$ degree integer polynomial. Let $q(x) = (p(x))^2 - 25$. Prove that there are not more than $2003$ distinct integers $m$, Such That $q(m) = 0$
To prove this, I assumed the contrary, and then factorised $q(x) = (p(x)+5)(p(x)-5)$. Then, by the pigeonhole principle, at least one of $p(x)+5, p(x)-5$ has at least 1002 distinct integer roots. We split into two cases, where one has $p(x)-5$ having at least $1002$ roots, and the other ...
Case 1: $p(x)-5$ has at least $1002$ roots. Let these roots be $r_1, r_{2}...$. Then we may write
$$p(x)-5=s(x)(x-r_1)(x-r_2)...(x-r_{1002})$$ It is easily verified that s(x) has rational coefficientss. However, I could not solve this problem until I verify that s(x) has integer coefficients. I have done some research, and found that perhaps Gauss' Lemma is able to prove this. However, I am not familiar with the lemma, and there are many different sources online which share a somewhat differing statement on the Lemma. I would appreciate it if someone could describe the Lemma, and perhaps use the lemma(or other methods) to prove if $s(x)$ is an integer polynomial.
