Mathematicians have a huge bag of tricks, which provide them with various ways of approaching a problem. In this note, we introduce an extremely useful lemma, which applies to polynomials with integer coefficients.
Lemma. If is a polynomial with integer coefficients, then for any integers and , we have
1) Show that for any integers and , we have
2) Using the above, prove the lemma.
3) Come up with a (essentially) two-line solution to question 4 from the book.
Note: In Help Wanted, Krishna Ar mentioned that problem 4 from the book was hard to solve. If you looked at the solution, it was also hard to understand what exactly was happening. This provides us with a simple way of comprehending why there are no solutions.
4) Come up with an alternative solution to E6 (page 250, from the book) using this lemma.
5) * (Reid Barton) Suppose that is a polynomial with integer coefficients. Let be an odd positive integer. Let be a sequence of integers such that
Show that all the are equal.
Where did you use the fact that is an odd positive integer?