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{\bf{\centerline{ Math 431 Problem Set 1}}}
``Ex a.b'' refer to exercise a.b in the text ``A friendly introduction to number
theory'' by Silverman. We say that an integer $d$ is a divisor of an integer
$n$, written $d\vert n$, if $n=kd$ for some integer $k$.
\medskip

\bigskip
\parindent=0pt{\bf{1.}} Find some Pythagorean triples.\smallskip
{\bf 2.} Find all Pythagorean triples $(a,b,c)$ with $c\leq 40$.\smallskip 
\smallskip  {\bf{3.}} Ex 1.2\qquad 
{\bf{4.}} Ex 1.3\qquad {\bf{5.}} Ex 2.1 \qquad
{\bf{6.}} Ex 2.2 \qquad {\bf{7.}} Ex 2.3 \qquad
{\bf{8.}} Ex 3.1 \qquad
\smallskip

{\bf Numerical Problems}\smallskip 

{\bf 9.} The common divisors of $30$ and $24$ are $1$,
$2$,
$3$,
$6$. The largest of these is $6$. What is the relationship between the largest
common divisor and all the other common divisors? 
Do the analogous problem for multiples instead
of divisors.\smallskip

{\bf 10.} Write 27319 in base $2$; in base $7$.\smallskip

{\bf 11.} Without changing to base 10, write (a)  $(1256)_8$ in base $2$
(b) $(1101101011)_2$ in base $8$; in base 16 (you may need to use digits $A$,
$B$, $C$, $D$, $E$, $F$ for $10$, $11$, $12$, $13$, $14$, $15$).\smallskip

{\bf 12.} Without changing to base 10, (a) add $(6124)_7$ and $(3145)_7$
(b) multiply them (c) subtract them (d) divide $(6124)_7$ by $(56)_7$.\smallskip

{\bf 13.} (a) Write $1/3$ and  $1/7$ in base $2$; in base $8$.
(b) Write as much of the octal (base $8$) expansion of $\pi$ as you can given
that the first $6$  digits in the decimal expansion of $\pi$ are
$3.14159$.\smallskip

{\bf 14.} What is the meaning of 
\quad $\sum_{d\vert n, d>0}
d$?\qquad of\quad
$\sum_{d\vert n, d>0} 1/d$?\qquad of\quad $\sum_{d\vert n, d>0} 1$? Calculate
them for
$n=6$,
$n=28$, $n=496$.\smallskip 

{\bf Exploration} \smallskip
{\bf 15.} What is the rule used to generate the sequence $$7,
22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,4,2,1,\ldots$$ from the starting value
$7$? Try generating a sequence the same way with different
 starting values. Any conjectures?\smallskip

{\bf Ingenuity} \smallskip
{\bf 16.} (a) Prove that $\pi$ (the circumference of a circle of diameter $1$) is
not $3$. (b) Prove that $\pi\neq 22/7$.\smallskip
{\bf 16.} Prove that the product of four consecutive positive integers is never a
perfect square. 

 
\bye