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\def\Z{{\mathbf Z}}
\def\Q{{\mathbf Q}}

\centerline{\bf Mathematics 361:  Number Theory}
\centerline{\bf Assignment B}

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{\bf Reading:  }Ireland and Rosen, Chapter 2 (including the exercises)
and into Chapter~3

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{\bf Problems:}

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{\em Even perfect numbers (nobody knows if there are any odd ones).}

1. The {\em sum of divisors\/} arithmetic function
$\sigma(n)=\sum_{0<d\mid n}d$ is introduced in Ireland and Rosen, and
$\sigma(p^a)=(p^{a+1}-1)/(p-1)$ by the finite geometric sum formula,
and $\sigma$ is multiplicative.  A positive integer is called
{\em perfect\/} if it is the sum of its proper positive divisors,
i.e., if $\sigma(n)=2n$.

(a) Show that if $2^p-1$ is prime (forcing $p$ to be prime) then 
$2^{p-1}(2^p-1)$ is perfect.

(b) If $m$ is even and perfect, show that $m$ takes the form
$m=2^{p-1}(2^p-1)$ where $2^p-1$ is prime.  (Write $m=2^{p-1}t$ where
$p\ge2$ and $t$ is odd and we don't yet know whether $p$ is prime.
Show that $\sigma(t)=2^pr$ where $r$ is odd and $t=(2^p-1)r$.  Note
that $r$ and~$t$ are distinct factors of~$t$.  Use this to show that
$r=1$ and $2^p-1$ is prime.)

(c) Where does the argument in (b) break down if $p=1$?  That is, why
can't we argue as in (b) to show that there are no odd perfect
numbers?

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2.  Work Ireland and Rosen exercises 1.30, some of 1.32--1.38.

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3.  Prove yet again that there are infinitely many primes by working
Ireland and Rosen exercise~2.3 or exercises~2.4--2.5 or
exercises~2.6--2.8.  

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4.  Get some practice with arithmetic functions by working a selection
from Ireland and Rosen exercises~2.9--2.21.

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5.  Acquire some initial familiarity with the Riemann zeta function of 
analytic number theory by reading and perhaps working Ireland and Rosen
exercises 2.25--2.26.  Also, 2.27 is a nice variant of Euler's
argument; to get going on it, note that every positive integer~$n$
factors uniquely as $n=m^2r$ with $r$ squarefree.

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