## A Bijection

Posted in Math by pgadey on 2013/03/30

While grading an assignment on cardinality, I ran into the answer to the following problem:

Exercise 1 Show that ${f(n) = \sum_{k=0}^n (-1)^{k+1} k}$ is a bijective map ${{\mathbb N} \cup \{0\} \rightarrow {\mathbb Z}}$.

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## Polynomials

Posted in Math by pgadey on 2013/03/01

Brandon Hanson told me the following elementary number theory problems last night.

Exercise 1 Every non-constant polynomial takes on a composite value.

Hint: Look at ${f(x) = p}$ and ${f(kp + x)}$.

Exercise 2 If a non-constant polynomial takes on infinitely many prime values then it is irreducible.