is a relation that will hold for any polynomial of degree 2. It is interesting as a side topic to be able to generate this kind of simple relationships, not just for coefficients 1 and -1, but for small coefficients. This can be resolved with signatures of expressions and the LLL lattice reduction algorithm, and it is a very good example of both. The problem can be formalized as follows: we are looking for a polynomial

For this we will construct the
matrix

where

For example, if we let *d*=2,
*n*=100000007, *m*=6 and *b*=10 and
the random variables assigned to the unknowns:
and
*x*=56110369 then the matrix is

Applying the LLL algorithm to the above produces the reduced matrix:

From these short vectors we can conclude the previous relation from row 3 and the relation

2*p*(*x*+1)+*p*(*x*+4) = *p*(*x*)+2*p*(*x*+3)

from row 1. These relations can be verified with a computer algebra system if necessary.