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{\bf{\centerline{ Math 431 Problem Set 2}}}


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{\bf Figurate numbers}\smallskip

{\bf1.} Write down the first ten triangular numbers, squares,
pentagonal numbers, hexagonal numbers. Write down the sum of the
first $k$ cubes, from $k=1$ to $k=10$.\smallskip 

{\bf 2.} Find a formula for the $n$-th triangular
number, the $n$-th pentagonal number, the $n$-th hexagonal number,
 the $n$-th $k$-gonal number. \smallskip

 
\smallskip  {\bf{3.}} Find and prove a relationship between the square numbers
and the sums of two consecutive triangular numbers.\smallskip
{\bf 4.} Find and prove  a relationship between triangular numbers,
square numbers and
the sum of the first $k$ cubes.\smallskip
{\bf 5.} 
Is three  times a pentagonal number always a triangular number?\smallskip

{\bf 6.} Determine which hexagonal numbers
are also triangular  numbers. \smallskip
{\bf 7.} Ex 1.1 The first two numbers which are both squares and triangles are
$1$ and
$36$. Find the next one, and if possible, the one after that. Can you figure out
a way to efficiently find square-triangular numbers? Do you think that there are
infinitely many?

 \smallskip

{\bf Numerical Problems}\smallskip

{\bf 8.}  Write down some of the
multiples of
$6$ and some of the multiples of
$8$. Determine which integers can be expressed as $6x+8y$ for integers $x$, $y$.
\smallskip 
{\bf 9.} Ex 6.1: 



{\bf 10.} Ex 6.2:

{\bf Exploration:  Continued fractions and Euclid's Algorithm} \smallskip
{\bf 11.} (a) Consider the ``magic table'' 
$$\matrix{\ &\ &2 &1&5&2\cr
     0&1&2&3&17&37\cr
      1&0&1&1&6&13}$$
     
       Calculate the determinants of all the $2\times 2$-matrices
consisting of the entries in two consecutive columns of the two bottom rows.
What do you notice?  Can you prove it holds for any magic table?
Write down integers $x$, $y$ satisfying $37x+13y=1$.



(b) Use Euclids algorithm to find the gcd of $37$ and $13$.
Use the results of this algorithm to show that 

$$37/13=2+{1\over{1+{1\over{5+{1\over 2}}}}}$$
(this is called a simple continued fraction for $37/13$).
 The fractions  $2+{1\over{1+{1\over{5}}}}$, $2+{1\over{1}}$, $2$
   are 
called the successive convergents to the fraction $37/13$. Can you explain
the relationship of the convergents to the magic table in (a)?
For each convergent $a/b$, calculate $37/13-a/b$. What do you notice?
Can you prove it in general?\medskip

{\bf 12. Egyptian fractions} The ancient Egyptians had symbols only for the
unit fractions i.e. those with numerator $1$, so they represented other fractions
as sums of unit fractions. The following procedure for writing a fraction in this
way was apparently known to them.

  Choose a rational number
$\alpha_0$ with
$0<\alpha_0<1$. Let $n_1$ be the smallest positive integer such that
$\alpha_1:=
\alpha_0-{1\over{n_1}}\geq 0$. If $\alpha_1>0$, let $n_2$ be the smallest
positive integer such that  $\alpha_2:=
\alpha_1-{1\over{n_2}}\geq 0$. Continue in this way to define $\alpha_1, 
\alpha_2, \alpha_3,\ldots $, as long as the $\alpha$'s are non-zero.
For example, if $\alpha_0=4/5$, then $n_1=2$, $\alpha_1=4/5-1/2=3/10$, $n_2=4$
and $\alpha_2=3/10-1/4=1/20$, $n_3=20$, $\alpha_3=0$ and the process stops. 
Note this represents $4/5=1/2+1/4+1/20$ as a sum of unit fractions.
\smallskip
(a) Calculate the $\alpha_i$ beginning with some other values of $\alpha_0$.
Does the process always seem to stop? If so, look carefully at your examples and
try to explain why it stops.
\smallskip
(b) Use Mathematica or Maple to calculate the $\alpha_i$ for the following
values of
$\alpha_0$:  
$5/121$, $65/131$, $31/131$.    









%{\bf Ingenuity} \smallskip


 
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