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We investigate the divisors dd of the numbers P(n)P(n) for various polynomials P∈Z[x]P∈ℤ[x] such that d≡1(modn)d≡1(modn). We obtain the complete classification of such divisors for a class of polynomials, in particular for P(x)=x4+1P(x)=x4+1. We also construct a fast algorithm which provides all such factorizations up to a given limit for another class, for example for P(x)=2x4+1P(x)=2x4+1. We use these results to find all the divisors d=2mk+1d=2mk+1 of numbers 24m+124m+1 and 24m+1+124m+1+1. For the numbers 24m+124m+1 the complete classification of such divisors is provided while for the numbers 24m+1+124m+1+1 the given classification is proved to be exhaustive only for m≤1000m≤1000

Finding special factors of values of polynomials at integer points