[FOM] Prime values of polynomials

[Harvey Friedman]

On Feb 28, 2008, at 4:24 PM, JoeShipman at aol.com wrote:

>
> I have read that no integer-coefficient polynomials of degree >1 are
> known to take infinitely many prime values; conjecturally, all the
> irreducible ones do.
>
> This is a nice example, but it's not so easy to tell whether a
> polynomial is irreducible.
>
> Can anyone provide a comparably simple example of a property which is
> believed to hold for all integers, but which is not known to hold for
> any? (Alternatively, I'll accept an example of a property which is
> conjectured to hold for all members of a set X but is not known to  
> hold
> for any, where X is easier to recognize than "irreducible polynomials
> of degree >1" even if X is not as easy to recognize as the integers
> are).
>
How about this property of k > 0:

There are infinitely many primes p such that p+2k is prime.

If you want the property to be of all integers, take:

There are infinitely many primes p such that p+4k+2 is prime.

I doubt if it is known that there exists k for which this holds (in  
both examples). This is stronger than saying that it is not known, for  
any specific k.

For another kind of example, consider the following property of k > 0:

e^k + (pi)^k is irrational.

Incidentally, if I proposed

k(e + pi) is irrational

then you would not like this. I think you want the additional  
constraint that no relation between the various instances is known.  
Perhaps you want to weaken that constraint?

Harvey Friedman