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\centerline{\bf Comments on }
\centerline{\bf Gasarch Primes 8-7-26} 

I will present a big structural correction that involves
re-arranging things, and then also some corrections on what
you actually wrote. DO the corrections on what you actually wrote
FIRST since once you change things around it may be hard to
find out which part I was pointing to. 

\begin{enumerate}
\item
Page 1. Title. You need a better title. This goes along with
the next comment.

\item
Overall Comment.
The theme should Elementary proofs that subsets of the primes are infinite.
that could also be the title

\centerline{\bf Elementary Proofs that Subsets of the Primes and Infinite}

With that in mind, the paper should be structured as follows

\begin{enumerate}
\item
{\bf Introduction}

Your first paragraph should be something like the following:

{\it By Dirichlet's theorem: 

For all $a,b$, $gcd(a,b)=1$, the set 

$$\{ p \colon p\hbox{ is prime and } p\equiv a \pmod b \}$$

is infinite. 

This proof is difficult. In this paper we will 
provide elementary proofs of subcases. }

\item 
{\bf Definitions and Examples}

Define Integral Domain, units, primes, irreducibles, UFD, Euclidean domains.

Give examples and counterexamples. 

Examples of primes. Problem 7, 8 and also do $\Z[i]$, $\Z[\sqrt{2}]$. 

Examples units- Problem 7, 8, use $\Z[i]$ and $\Z[\sqrt{2}]$

Example of UFD and NON-UFD (Problems 2 and 3)

Examples of when primes and irred are the same and when they are not. 

\item
{\bf The Set of Primes $\equiv a \pmod 4$ where $a\in\{1,4\}$ is Infinite}

\item
{\bf The Set of Primes $\equiv a \pmod 5$ where $a\in\{1,2,3,4\}$ is Infinite}

\item
{\bf The Set of Primes $\equiv a \pmod 6$ where $a\in\{1,5\}$ is Infinite}

\item
{\bf Limitations to our Methods}

Discuss the difficulties you have getting a result for mod 7. 

\end{enumerate}

\item
Page 1.
Lemma 0.1. The word {\it multiples} means multiples by an integer.
You should clarify that its multiples by an elements of $\Z[\sqrt{-k}]$. 

\item
Page 2. 
Lemma 0.4. You should say that $\Z[\sqrt{-k}]$ is a Euclidean Domain
USING the norm $N$. 

\item 
Page 2. 
In your note after the proof of Lemma 0.4 you should clarify that for
$k\ge 3$ the norm would not work. Also see next point.

\item
Page 2. 
Your note after the proof of Lemma 0.4 only shows that THIS norm would not work.
How about another norm? Do you have a proof that if $k\ge 3$ then
$\Z[\sqrt{-k}]$ is NOT a Euclidean domain--if thats true.
You later show that $\Z[\sqrt{-5}]$ is not a UFD and hence is not an ED.
But what about (say) $\Z[\sqrt{-7}]$?

\item
Page 3. You use the theorem that if $p\equiv 1 \pmod 4$ then $-1$ is a square.
Thats fine. But you should make that into a Lemma (which you don't have to prove)
and then refer to the Lemma in your proof. 

\item 
Page 3. Problem 5. there is no space between mod and 5. 

\item
Page 3. Problem 5. The title indicates that you will show that

{\bf The Set of Primes $\equiv a \pmod 5$ where $a\in\{1,2,3,4\}$ is Infinite}

but the section seems to only prove this for $a=1,4$. 

\item
I have not read the proofs carefully. I will do that for the next version you
send me. 

\end{enumerate}

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