
My proof that "a polynomial of single variable with integer coefficients cannot generate primes for all integer inputs"
Posted:
Aug 19, 2013 11:21 PM


So I came across f(n) = n^2  n + 41 in this video: http://www.youtube.com/watch?v=3K12i0jclM
So n=41 clearly will produce a composite number. So based on this fact, I try to do my proof at the very bottom...
So it seems that no polynomial in a single variable (I guess we have to through out constant polynomials) exists that only produces primes on integer input. And wolfram link claims someone proved this in 1752 (http://mathworld.wolfram.com/PrimeGeneratingPolynomial.html).
Now it "seemed" obviously easy to prove based on the example at the top, because:
Let f(n) be a nonconstant polynomial with integer coefficients.
case 1) if the polynomial has no constant term, then any n would divide f(n) so let n=2 and we are done
case 2) if the polynomial has a constant term then, the constant would divide the polynomial evaluated at the constant, i.e., c divides f(c)
but this proof does not work for when the constant=1. 1 dividing f(1) does not prove that f(1) is composite.
How do I reconcile this?

