# Many descendants

Algebra Level pending

Let $$p$$ is a prime number. Denote by $$I(p)$$ the number of irreducible monic polynomials with coefficients in $${\mathbb Z/pZ}$$ (integers modulo $$p$$) that divide the polynomial $$x^{p+1}-x+1.$$

For $$p<100?$$, what is the largest possible value of $$I(p)$$ ?

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial $$x^3 + 3x - 5$$ is monic but the polynomial $$-x^4 + 2x^3 - 6$$ is not.

A polynomial with coefficients in $${\mathbb Z/pZ}$$ is irreducible if it cannot be expressed as a product of two non-constant polynomials with coefficients in $${\mathbb Z/pZ}.$$

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