Naturally, one might wonder whether each of these more specific list of prime numbers is infinite. As it turns out, the Euclidean argument can indeed be extended to each of these cases. Specifically, in the case of primes congruent to 3 modulo 4, the proof is a direct extension of Euclid's original argument.
Naturally, one might wonder whether each of these more specific list of prime numbers is infinite. As it turns out, the Euclidean argument can indeed be applied to each of these cases. Specifically, in the case of primes congruent to 3 modulo 4, the proof is a direct extension of Euclid's original argument.
There are infinitely many primes that are congruent to 3 modulo 4.
\end{theorem}
@ -66,7 +66,7 @@
\begin{theorem}[\cite{MurtyandThain}]
A "Euclidean proof" exists for the arithmetic progression l (mod k) if and only if $l^2\equiv1$ (mod k).
\end{theorem}
Thus, Euclid's argument, when applied to the arithmetic progression of prime numbers, has been exhausted. We will prove the "if" direction of Theorem 3.2 by following the outline of the argument given by Murty and Thane modulo some field theoretic results which are deemed outside the scope of this project.
Thus, Euclid's argument, when applied to the arithmetic progression of prime numbers, has been exhausted. In our subsequent research, we will prove the "if" direction of Theorem 3.2 by following the outline of the argument given by Murty and Thane modulo some field theoretic results which are deemed outside the scope of this project. This proof will also require the inclusion of more results which extend the Euclidean argument to polynomials. Finally, we will provide a more complex "Euclidean proof" of an arithmetic progression of prime numbers that is guaranteed to exist by Theorem 3.2.
\begin{thebibliography}{2}
\bibitem{Silverman} Joseph Silverman, \textit{A Friendly Introduction to Number Theory} p.83-140, Pearson, Boston, 4th edition, 2011.
