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{\bf{\centerline{ Math 431 Final Exam : Wednesday, May 10, 4:15--6:15pm}}}


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{\bf 1.} {\sl (20 points)}\smallskip
{(a)}  Find all incongruent solutions of the congruence $36x\equiv 44
\pmod{88}$.\smallskip
{(b)} Find the  two rightmost digits of the base ten expression of the number 
$3^{1000}$.\medskip 
{\bf 2.} {\sl (30 points)}\smallskip
{(a)} Show that $a=21\equiv -2$ is a primitive root modulo $p=23$
.\smallskip  
{(b)} Construct a table of indices modulo $23$ with $a$ as base. \smallskip
{(c)} Use the table of indices from (b) to determine all incongruent  solutions
of the congruence $5x^{19}\equiv 7\pmod{23}$.\medskip
{\bf 3.} {\sl (20 points)}\smallskip
{(a)} Determine whether $51$ is a quadratic residue modulo the prime $103$.
\smallskip
{(b)} Use the law of quadratic reciprocity to describe for which primes $p$ 
 the number $5$ is a quadratic residue modulo $p$.
\medskip{\bf 4.} {\sl (20 points)}\smallskip
 {(a)} The numbers $p=1019$ and $q=1021$  are
primes. Determine which of the following integers are expressible as a sum of two squares: 
$p$, $q$, $p^3q^5$, $p^2q^5$ and $p^3q^2$.\smallskip
{(b)} Find   a Pythagorean triple $(a,b,c)$ of integers, $c^2=a^2+b^2$, with 
$c=41$. Is there  a Pythagorean triple $(a,b,c)$ with $c=43$? Explain.  \medskip
{\bf 5.} {\sl (30 points)}\smallskip 
(a) Use the line through the points $(-2,1)$ and $(1,2)$ on the elliptic curve
$E:y^2=x^3-2x+5$ to produce a third rational point on $E$.\smallskip(b) Consider the following
proceedure: take a  point $P=(a,b)$ on $E$, find by calculus the equation of the tangent line $T$ to
$E$ at $P$, and let $Q$ denote the  point of intersection of $T$ with $E$ satisfying $Q\neq P$.
 Find
$Q$ if  $P=(-2,1)$. Try to explain why this proceedure always gives a rational point $Q$ when
$P$ is  rational. 
\smallskip (c) Calculate the number of points  on the elliptic curve $E \pmod 5$.\smallskip
\medskip
 
{\bf 6.} {\sl (30 points)} The cubes modulo
$7$ are
$0$,
$1$ and
$6$ since
$0^3\equiv 0$, $1^3\equiv 2^3\equiv 4^3\equiv 1\pmod 7$ and $3^3\equiv 5^3\equiv 6^3\equiv 6\pmod
7$. Let $c_p$ denote the number of cubes modulo $p$ i.e. $c_p$ is the number
of distinct values of $x^3\pmod p$. So, for example, $c_7=3$. \smallskip {(a)} 
Calculate $c_p$ for $p=2$, $3$, $5$, $7$, $ 11$ and $13$.  
\smallskip {(b)} 
Try to guess a formula for $c_p$, for all prime numbers $p\neq 3$.
(Your answer should have two cases, depending on whether $p\equiv 1\pmod 3$ or $p\equiv 2\pmod
3$. You may need to compute $c_p$ for more primes.)
\smallskip {(c)} Try to prove that your formula in (b) is correct. \medskip
  


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