Question

Let f ∈ Z[x] be a nonconstant polynomial with the property that all the roots (in comlex plane) for the equation f(x) = 0 are distinct. Prove that there exist infinitely many positive integers n such that f(n) is not a perfect square.

Answer 1

Let C  be the algebraic variety defined (over Q) by the equation y²=f(x)

Under the assumption that f has different roots (in fact, a weaker assumption that f is not a square of another polynomial will suffice), we know that C defines a curve (i.e. it is irreducible). It is birationally equivalent to a unique projective smooth curve C′, which has a genus g= floor of d−12, where d is the degree of function f.

Now we may apply Siegel's theorem: if g>0 then there are only finitely many integral points on C, hence not all f(n) can be squares.