AoPS Academy
be real numbers such that![\[a+b+c+d=-2\]](http://latex.artofproblemsolving.com/c/7/3/c73dff68dcdfbe4f6b6b2f52b378548e47193a16.png)
![\[ab+ac+ad+bc+bd+cd=0\]](http://latex.artofproblemsolving.com/9/f/4/9f450dcb150229ee76e3debdaceadddb4e451458.png)
is not greater than 
.
define 
be the product of first 
primes. Determine if there exists positive integers 
with the following property :![\[ m?=n(n+1)(n+2)(n+3) \]](http://latex.artofproblemsolving.com/4/5/c/45cb4a00f9395b8c6edb08a343c3bbd664916a4d.png)
and distinct positive integers 
are given. Does there exist a polynomial ![$p(x)\in\Bbb{Z}[x]$](http://latex.artofproblemsolving.com/5/1/a/51a23efd70e495d2fe516b642f0a6e71e5875ba2.png)
of degree 
that satisfies the following conditions?
of positive integers such that
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
and 
is a solution and 
. Then 
is new pair. Also 
and 
. We get infinitely many pairs.
