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\centerline{\bf Comments on }
\centerline{\bf Proving Pi is not Equal to Various Fractions}

\begin{enumerate}
\item
Page 1. Title. The Pi in the title should be $\pi$. 

\item
I should be listed after the authors as your mentor. 

\item
Page 1. Once you have that $\pi$ is irrational, you do not need to
show that its transcendental to establish its not rational. You should
be making the following points

It is know that $\pi$ is irrational, and hence not any of the three
fractions we are considering. 

The proof that $\pi$ is irrational is difficult so we are inspired to see
if we could prove $\pi$ is not any of those fractions using elementary means.

Incidentally, $\pi$ is not just irrational, its actually transcendental. 

\item
Page 1 and 2- to get $PI(n) < \pi < PC(n)$ you need that the distance from
the center to any vertex is exactly 1. (Check that.)

\item
The table. $X(n)$ should also be on the table and you should
speculate on what $X(n)$ is. This is true for the later tables
as well, but I won't mention it again.

\item 
Page 4. You have {\it When solving the Basel problem Euler proved that}

Finding that sum IS the Basel problem.

Here is what you can say:

{\it In 1644 Pietro Mengoli posed the following problem: Find the exact
value of $\sum_{i=1}^\infty \frac{1}{i^2}$. This was was called
{\rm The Basel Problem} because Euler and the Bernoulli family worked on
it, and they lived in Basel. Finally Euler solved it. Euler solved it
in 1735 by showing that

$$\sum_{i=1}^\infty \frac{1}{i^2}=\frac{\pi^2}{6}.$$

}

\item 
You should make it clear that 

$$\pi = \sqrt{6\sum_{i=1}^\infty  \frac{1}{i^2}}$$

and hence, for all $n$, 

$$\pi > \sqrt{6\sum_{i=1}^n \frac{1}{i^2}}$$

and then define $L(n)$. 

\item 
The following is very interesting and may lead to more questions:

\begin{itemize}
\item
For the inscribe-circumscribe method 

$333/106$ needed $n=249$

$355/113$ needed $n=6224$


\item
For the Basel method 

$333/106$ needed $n=11475$

$355/113$ needed $n=1337$

\end{itemize}

SO using ins-circum method 333/106 is easier than 355/113
but using Basel 355/113 is easier than 333/106.

You should point this out. 

\item
At the end of the paper you should have a table of
all three fractions and all the methods, and the least $n$
that shows they are not $\pi$. 

\item
Its interesting that the Nilakantha series has so much smaller values
of $n$ then the first two methods. Point that out.

\end{enumerate}
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