# 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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