Factor A Degree 100000000 Polynomial

Algebra Level 5

We are considering polynomials of the form \[ (x-1)(x-2)(x-3) \cdots (x-p +1) + 1 -p^2\] that can be factorized into 2 non-constant polynomials with integer coefficients.

What is the number of those polynomials if p is prime and less than \(10^8\)?


Inspiration.

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