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Topic: My proof that "a polynomial of single variable with integer
coefficients cannot generate primes for all integer inputs"

Replies: 6   Last Post: Aug 20, 2013 3:43 PM

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 gaya.patel@gmail.com Posts: 160 Registered: 11/29/05
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
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So I came across f(n) = n^2 - n + 41 in this video:
http://www.youtube.com/watch?v=3K-12i0jclM

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/Prime-GeneratingPolynomial.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?

Date Subject Author
8/19/13 gaya.patel@gmail.com
8/20/13 quasi
8/20/13 quasi
8/20/13 Peter Percival
8/20/13 quasi
8/20/13 Peter Percival
8/20/13 quasi

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