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 2 of 4: Count roots of one level equation
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d=deg⁡(P)d=\deg(P)
Detailed analysis

The integers satisfying P(k)=1 and those satisfying P(k)=-1 are disjoint. Each equation is a nonzero polynomial equation of degree d, so either one level set by itself contains at most d integer roots.