\[P(x)\]
that returns only prime numbers. If the polynomial factorised, then obviously all would return a composite number for large enough \[x\]
but polynomials that do not factorise do not return only prime numbers.\[P(x)=x^2+x+1\]
returns a composite number for \[x=7\]
(\[P(4)=4^2+4+1=21=3 \times 7\]
).\[P(x)\]
does factorise modulo 7 however.\[x^2+x+1 \equiv x^2+8x+15 \; (mod \; 7) \equiv (x+3)(x+5) \; (mod \; 7)\]
We can force
\[P(x)\]
to be composite by choosing \[x\]
so that one of the factors is zero modulo 7, then \[P(x)\]
is composite.