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{\bf{\centerline{ Math 431 Problem Set 4}}}


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{\bf }\smallskip

{\bf 1.} Ex 9.1 \smallskip

{\bf 2.} Ex 9.2\smallskip 

{\bf 3.} Ex 10.1 
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\smallskip  {\bf{4.}} Ex 11.1 \bigskip



 {\bf 5.} Ex 11.2 
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{\bf 6.} Ex11.3 
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{\bf 7.} Ex 11.4

\bigskip {\bf Exploration.}
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{\bf 8. Sums of powers modulo a prime.}  
Let $p$ be an odd prime, $k\geq 1$ and $S=\sum_{j=1}^{p-1}j^k$.
Compute the values of $S\pmod p$ for some $p$ and $k$ and try to 
form a conjecture on the value of $S\pmod p$. Can you prove the conjecture, or
part of it?
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{\bf 9. Wilson's theorem for non-primes.}
 Calculate the value of $(m-1)!\pmod m$ for various composite numbers
$m$. Formulate a conjecture on its value and prove your conjecture. 

{\bf 10.} Find the number's of the solutions of the congruences $x^2-1\equiv 0 $
$\pmod 3$, $\pmod 8$, $\pmod {24}$. Same for $x^2-x-2$. What do you notice.
Explain your observations. Explain how to systematically use the lists of
solutions of these congruences $\pmod 3$ and $\pmod 8$ to find all solutions
$\pmod {24}$.\smallskip

{\bf 11.} Find the largest power of $2$ exactly dividing 
$1000!$. Find a formula for the largest power of a prime $p$ dividing $n!$. Find
the prime factorization of
$100!$, of
${100\choose 40}={{100!}\over {40!60!}}$.\smallskip

{\bf 12.} Calculate $\sum_{d\vert n,d>0} \phi(d)$ for various values of $n$.
What do you notice? Can you prove it?

{\bf 13. Continued fractions.}  
Consider a fraction of the form 
 $$q_1+{1\over\displaystyle{q_2+{1\over\displaystyle{q_3+\ldots{+
\displaystyle{1\over q_n}}}}}}$$ in which $q_1$, $q_2$, $\ldots$, $q_n$ are
not necessarily integers. For brevity, write such a fraction in the form
$[q_1;q_2,\ldots, q_n]$. Set $P_{-1}=0$, $P_0=1$, $P_j=P_{j-1}q_j+P_{j-2}$ and 
$Q_{-1}=1$, $Q_0=0$, $Q_j=Q_{j-1}q_j+Q_{j-2}$.
Show that 

(a) $[q_1;q_2,\ldots, q_j]=P_j/Q_j$ for $j=1$, $2$, $\ldots$, $n$.

(b) $P_{j-1}Q_j-Q_{j-1}P_j=(-1)^{j-1}$ for $1\leq j\leq n$.

(c) $P_{j-2}Q_j-Q_{j-2}P_j=(-1)^{j}q_j$ for $2\leq j\leq n$.

What does this have to do with magic tables? \smallskip










{\bf Ingenuity}\smallskip 

{\bf 14.} Show that the sum $1+{1\over 2} +{1\over 3}+\ldots +{1\over n}$ is
never an integer for $n\geq 2$.

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