MathLabs

Problem 6

Let PP be a non-constant polynomial with integer coefficients. If n(P)n(P) is the number of distinct integers kk such that P(k)2=1P(k)^2=1, prove that n(P)−deg⁡(P)≤2n(P)-\deg(P)\le2.
Step 3 of 4: Start from the smallest exceptional integer
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r=min⁡{k:P(k)2=1}r=\min\{k:P(k)^2=1\}
Detailed analysis

Let r be the smallest integer with P(r)=1 or -1, and count the roots of the corresponding equation. There are at most d of them. Every root of the other equation is at least r and cannot equal r; the divisibility result therefore restricts it to r+1 or r+2.