# Interesting question

David Wilson dwilson at gambitcomm.com
Mon Aug 18 18:13:24 CEST 2008

```I don't know if there's a theorem to that effect. I do know that there
are polynomials in several variables whose positive values are the
primes (but I'm not sure what the conditions are on the variable
values). I don't know if there is a known nontrivial lower limit on the
number of necessary variables to achieve this.

Anyway, here we are talking about numbers NOT of the specified form,
which I think is a whole nother beastie (people actually say "a whole
nother" in preference to "a whole other"). At any rate, the observation
that xy+yz+zw+wx = (x+y)(z+w) makes it obvious that the expression takes
on composite values when x,y,z,w are positive integers. What's
interesting is that x1*x2 + x2*x3 + ... + x_n*x1 seems to be factorable
like this only for n = 4.

Jim Nastos wrote:
> Interesting indeed.... isn't there some theorem that says that no
> polynomial function can generate only primes? ... Or maybe that's a
> single-variable polynomial.
> JN
>
> On 8/18/08, David Wilson <dwilson at gambitcomm.com> wrote:
>
>> What positive integers are of the form wx+xy+yz+zw where w,x,y,z are
>> positive integers?
>>
>>
>>
>
>
>

```