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 1 of 4: Compare roots of the two level equations
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P(r)=1,P(s)=−1⟹s−r∣2P(r)=1,\quad P(s)=-1\quad\Longrightarrow\quad s-r\mid2
Detailed analysis

For integers r,s with P(r)=1 and P(s)=-1, the integer s-r divides P(s)-P(r)=-2, by the integer-coefficient divisibility property. Thus any such two roots differ by 1 or 2 in absolute value.