A shift and reduce

For how many positive integers \(N<1000\), does there exist a polynomial \(f(x)\) with integer coefficients such that all of the following conditions are true:

  1. The degree of \(f\) is at most \(6.\)

  2. The sum of absolute values of coefficients of \(f\) is at most \(7.\)

  3. The polynomial \(f(x)+N\) is reducible over the integers.

Details and assumptions

A polynomial that is reducible over the integers can be expressed as a product of two non-constant polynomials with integer coefficients.


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