From b8e7b6c22084d8f667f6c7d58d2a9617973e1241 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Fri, 12 May 2023 18:09:57 +0100
Subject: [PATCH] Revert "Modular arithmetic work actually irrelevant"
Realised I was wrong
This reverts commit 23826ea20433d5e78bafb5d447000a3cf6becc63.
---
main.tex | 22 ++++++++++++++++++++++
1 file changed, 22 insertions(+)
diff --git a/main.tex b/main.tex
index a812d2c..93741a6 100644
--- a/main.tex
+++ b/main.tex
@@ -838,6 +838,28 @@ this happens when:
< \frac{1}{\lcm(m,2n^2)}
\end{equation}
+\begin{sagesilent}
+rhs_numerator = (
+ positive_radius_condition
+ .rhs()
+ .subs([q_value_expr,beta_value_expr])
+ .factor()
+ .numerator()
+)
+\end{sagesilent}
+
+\noindent
+Considering the numerator of the right-hand-side:
+
+\begin{align}
+ \sage{rhs_numerator}
+ &\equiv (a(-a^{-1}b)+2b)a &\mod n
+\\
+ &\equiv ab &\mod n
+\end{align}
+
+And so, we also have $a(ar+2b) \equiv ab$ (mod $\lcm(m,2n^2)$).
+
\minorheading{Irrational $\beta$}
--
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