On a "Singular" Integration Technique of Poisson

H = 22: here the smallest n for which the case d = 1 can be applied is n = 204, yielding f (204) = 1732807009, whereas Theorem 1 proves the irreducibility with n = 30, which gives f (30) = 7 · 118543, and with more than forty other values of n less than 204. Theorem 1 can also be applied if all values f (n) for n in Z are divisible by a common factor d > 1. But in this case a straightforward transformation of f (x) into a polynomial without this property may be more advisable than the direct application of the theorem. For instance, d = 2 divides all values of the polynomial f (x) = x4 + 9x2 + 4, and the fact that f (17) = 2 · 43063 establishes its irreducibility. On the other hand, the substitution x → 2x transforms this polynomial into g(x) = 4x4 + 9x2 + 1, which is irreducible because g(7) = 2 · 5023.