PROBLEM OF THE MONTH
October 2002
Show that if
are all distinct integers,
then the polynomial
cannot be written as a product of two other (non-constant)
polynomials with integer coefficients.
There are several possible solutions to this problem. Here is one short way to
tackle it:
Suppose we can write the degree polynomial as
for some nonconstant polynomials and with integer coefficients. In particular
one of these two polynomials will have degree at most , say .
The polynomial takes only strictly positive values (being a perfect square ),
so without loss of generality we may assume that both and take only
strictly positive values for all real values of .
If we substitute now the integers for we get that
for all
,
and since both and have integer coefficients
it follows that
for all . Thus both polynomials and take the same value
for distinct values of , and since
it follows that the only possibility is that
.
Taking the derivative of
and substituting then for
leads to
In other words the degree polynomial
vanishes
at distinct values of and thus must be identically zero.
Hence is a constant polynomial, and by the above
we must have
Since and take only (positive) integral values for
all integral value of , it follows that
which is a contradiction since both and are
polynomials of degree .