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{\bf{\centerline{ Math 431 Exam 2: Wednesday, March 29, 3:00--4:15pm}}}


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{\bf 1.} {\sl (20 points)}\smallskip (a) Determine $a=\phi(1763)$  where $\phi$
denotes Euler's phi-function (note $1763=41\times 43$).
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(b) Find explicitly  a positive solution $u$ of $187u\equiv 1\pmod{a}$.\smallskip
(c) Explain how to use your answer to (b) to find a solution $x$ of  $x^{187}\equiv 341
\pmod{1763}$ (i.e. express a solution of the congruence in terms of $u$). Is the solution unique
modulo $1763$?
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{\bf 2.} {\sl (15 points)} Let $p$ be a fixed prime number. \smallskip (a) Let $a$ be an integer not divisible
by 
$p$. Explain what is meant by the exponent $e_p(a)$ of $a$ modulo $p$.
\smallskip (b) For which values of the integer $d$ is there an integer $a$ which is not divisible by $p$ and
satisfies $e_p(a)=d$? For such a $d$, how many solutions $\psi(d)$  for $a$ of the equation  $e_p(a)=d$ are
there (do not count values of $a$ which are congruent modulo   
$ p$ as distinct solutions)?
\smallskip (c) If $g$ is a primitive root modulo $p$, what is the value of $e_p(g)$?\medskip
{\bf 3.} {\sl (25 points) } \smallskip
(a) Show that $2$ is a primitive root modulo $11$.\smallskip
(b) For each $a$ not divisible by $11$, let $I(a)$ denote the index of $a$ with respect to the
primitive root
$2$ i.e. the solution $j$ with $1\leq j\leq 10$ of the equation $2^j\equiv a\pmod{11}$. Construct a
table showing for each $a$ with $1\leq a\leq 10$ the values of $I(a)$ and $e_{11}(a)$. 
\smallskip (c) Use the table of indices in (b) to find all incongruent solutions of the equation
$$5x^5\equiv 6\pmod{11}.$$ \medskip
{\bf 4.} {\sl (20 points)} Let $p$ be an odd prime number. 
\smallskip (a)   Let $a$, $b$ be  integers not divisible by $p$. Explain what
is meant by $\bigl({a\over p}\bigr)$ (Legendre symbol). What is the relationship between 
$\bigl({ab\over p}\bigr)$, 
$\bigl({a\over p}\bigr)$ and  $\bigl({b\over p}\bigr)$?\smallskip 
(b) State the law of quadratic reciprocity (your answer should express  $\bigl({{-1}\over p}\bigr)$ and  $\bigl({2\over p}\bigr)$ in
terms of $p$, and describe the relationship between  $\bigl({q\over p}\bigr)$ and  $\bigl({p\over q}\bigr)$ for another odd prime $q$). 
\medskip{\bf 5.} {\sl (20 points)} \smallskip (a) The number $1307$ is prime. Use the law of quadratic reciprocity to determine if the
congruence $x^2\equiv 690\pmod{1307}$ has a solution.\smallskip
(b) Use quadratic reciprocity to determine for which odd primes $p\neq 7$ the congruence $x^2\equiv
7\pmod{p}$ has a solution (your answer should depend only on the value of $p$ modulo $28$). 
   






























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