Research Paper Update

Include Theorem 2 of Murty and Thane's paper at the end of the research paper (to be organized later).

research_paper.tex

@@ -138,6 +138,25 @@
     \section{Conclusions}
     Euclid, when proving that there is an infinite number of prime numbers, formulated an argument whose utility remained untapped for many years. Throughout this paper, that utility has been demonstrated by proving the infinitude of different categories of prime numbers. Though the argument does have its limits, it provides both elegant and relatively simple proofs when compared to alternative methods, and at the same time demonstrates that proof techniques can be have many useful applications beyond their original use.
     
+    \section{Murty and Thane Theorem 2}
+    \begin{theorem}[\cite{MurtyandThain}]
+        If $f \in \mathbb{Z} [x]$ is non-constant, then f has infinitely many prime divisors.
+    \end{theorem}
+    \begin{proof}
+        To begin, notice that $f$ has at least one prime divisor, becuase $f(x)= \pm 1$ has a finite number of solutions.
+        Now suppose $f(0)=c \neq 0$ and that f has a finite number of prime divisors $p_1, p_2, \dots p_k$. Let $$Q=p_1p_2 \cdots p_k.$$ \
+        Note that $f(x)$ has the form $c_nx^n+c_{n-1}x^{n-1}+ \cdots +c_1x_1+c$ and
+        \begin{align*}
+            f(Qcx)&=c_n(Qcx)^n+c_{n-1}(Qcx)^{n-1}+ \cdots +c_1(Qcx)+c \\
+            &=c[c_n(Qx)^nc^{n-1}+c_{n-1}(Qcx)^{n-1}c^{n-2}+ \cdots +c_1Qx+1] \\
+            &=cg(x),
+        \end{align*}
+        where $g(x) \in \mathbb{Z}$ is of the form $1+c_1x+c_2x^2+ \cdots$. \par
+        Now, $Q|c_i$ in for each $c_i$ in $g$. Additionally, $g \in \mathbb{Z}[x]$ must have at least one prime divisor, $p$, by the argument presented above. \par
+        Finally, $p|g$ implies that $p|f$ and thus $p|Q$. However, this is impossible, because $p|Q$ also implies that $p|1$ (in order for $p$ to be a prime divisor of $g$), which is a contradiction. Therefore $f$ has infinitely many prime divisors.
+        
+    \end{proof}
+    
     \begin{thebibliography}{2}
         \bibitem{Silverman} Joseph Silverman, \textit{A Friendly Introduction to Number Theory} p.83-140, Pearson, Boston, 4th edition, 2011.
         \bibitem{MurtyandThain} M.R. Murty; N. Thain, \textit{Prime Numbers in Certain Arithmetic Progressions}, Funct. Approx. Comment. Math. 35 (2006), 249-259.