Interesting question

David Wilson dwilson at
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> wrote:
>> What positive integers are of the form wx+xy+yz+zw where w,x,y,z are
>> positive integers?

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