From: rusin@math.niu.edu (Dave Rusin)
Newsgroups: sci.math.research
Subject: Can you characterize these integers?
Date: 11 Oct 1997 06:11:52 GMT

Say an integer  n  has property  P  if there exist positive integers x,y,z,w
such that   n = 4 x y z - w   and   w | (y+z). For example,  n=29  has
property  P  (take  x=1, y=2, z=4, w=3).

The following seem to be true based on a computer check:
   (a) if  n  is a square, then  n  does not have property  P;  conversely,
   (b) if  n  does not have property  P, then  n  is a square, or else
	n= 288, 336, 4545,  ... (?)

Got proof?

A weaker form of (b), namely
   (c) if  n  does not have property  P, then  n  is composite
(i.e., all primes have property P ) is sufficient to prove a
well-known conjecture of Erdos, which I won't name for fear of
spoiling your creativity :-)

dave

==============================================================================
From: thomaso@best.com (Thomas Andrews)
Newsgroups: sci.math.research
Subject: Re: Can you characterize these integers?
Date: 11 Oct 1997 12:36:43 -0700

In article <61n5b8$k4d$1@gannett.math.niu.edu>,
Dave Rusin <rusin@math.niu.edu> wrote:
>Say an integer  n  has property  P  if there exist positive integers x,y,z,w
>such that   n = 4 x y z - w   and   w | (y+z). For example,  n=29  has
>property  P  (take  x=1, y=2, z=4, w=3).
>
>The following seem to be true based on a computer check:
>   (a) if  n  is a square, then  n  does not have property  P;  conversely,

This one is not hard to show.  Basically, write y+z=kw and eliminate
z from the equation:

        n^2 = 4 x y (kw-y) - w
            = 4xykw - 4xy^2 - w
Or
        n^2 = w ( 4xyk - 1 ) - 4xy^2     (*)

A quick application of the Jacobi symbol, J, shows that:

	J(-P,4PK-1)=-1

for all P and K (seperate cases for P odd and even.)

So, in particular:

	J(-x, 4xykw-1) = -1

and -x cannot be a square mod (4xykw-1).

This means that we cannot solve (*).
-- 
Thomas Andrews         thomaso@best.com         http://www.best.com/~thomaso/
    "Show me somebody who is always smiling, always cheerful, always
    optimistic, and I will show you somebody who hasn't the faintest
    idea what the heck is really going on." - Mike Royko
==============================================================================
From: Thomas Andrews <thomaso@best.com>
To: rusin@math.niu.edu
Subject: Re: Can you characterize these integers?
Date: Sat, 11 Oct 1997 11:43:46 -0700 (PDT)
Newsgroups: sci.math.research

In article <61n5b8$k4d$1@gannett.math.niu.edu> you write:
>Say an integer  n  has property  P  if there exist positive integers x,y,z,w
>such that   n = 4 x y z - w   and   w | (y+z). For example,  n=29  has
>property  P  (take  x=1, y=2, z=4, w=3).
>
>The following seem to be true based on a computer check:
>   (a) if  n  is a square, then  n  does not have property  P;  conversely,
>   (b) if  n  does not have property  P, then  n  is a square, or else
>	n= 288, 336, 4545,  ... (?)

For (a), I can prove:

    If there is a solution for some square, then there is a solution
    for some square with y,z, and w relatively prime

    If y and z are odd, there is no square produced unless
    w is a multiple of 8.

I not sure what happens when w is a multiple of 8.

Basically, I used Jacobi symbol Quadratic Reciprocity.

>
>Got proof?
>
>A weaker form of (b), namely
>   (c) if  n  does not have property  P, then  n  is composite
>(i.e., all primes have property P ) is sufficient to prove a
>well-known conjecture of Erdos, which I won't name for fear of
>spoiling your creativity :-)
>
>dave
>
>


-- 
Thomas Andrews         thomaso@best.com         http://www.best.com/~thomaso/
    "Show me somebody who is always smiling, always cheerful, always
    optimistic, and I will show you somebody who hasn't the faintest
    idea what the heck is really going on." - Mike Royko