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\ctln{\bf Math 445 Homework 1}

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\ctln{Due Wednesday, Sept. 10}

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\item{1.} (NZM, Problem 1.3.27) Show that if $n$ is {\it not} prime, then $n|(n-1)!$ .

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\item{2.} (NZM, Problem 1.3.31) Show that if $f(x)$ is a non-constant polynomial
with integer coefficients, then $f(n)$ \underbar{cannot} be prime for every $n\in {\Bbb N}$.

\item{}(Hint: If $f(n)=p$ is prime, show that for every $k\in {\Bbb N}$ we have $p|f(n+kp)$;
eventually $f(n+kp)$ is too big to \underbar{be} $p$ ...)

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\item{3.} (NZM, Problem 1.3.33) Show that for $n>1$, $n^4+n^2+1$ is {\it never} prime.

\item{}(Hint: $f(x)= x^4+x^2+1$ can be expressed as a product of quadratics; find the factorization!)

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\item{4.} Show that if $2^n-1$ is prime, then $n$ must be prime.

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