From c0b192f0771600a9aaa00f834005ab79177c2b5e Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Fri, 12 May 2023 18:07:10 +0100
Subject: [PATCH] Start the 'min' expressions
---
main.tex | 18 ++++++++++++++++++
1 file changed, 18 insertions(+)
diff --git a/main.tex b/main.tex
index 38dc744..a812d2c 100644
--- a/main.tex
+++ b/main.tex
@@ -819,6 +819,24 @@ radius of the pseudo-wall being positive
\frac{1}{2n^2}\ZZ
\end{equation}
+For each $r$, the smallest element of $\frac{1}{\lcm(m,2n^2)}\ZZ$ strictly larger
+than the lower bound here is exactly $\frac{1}{\lcm(m,2n^2)}$ greater.
+Therefore, if any of the two upper bounds come to within
+$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for $d$.
+
+Considering equations
+\ref{eqn:bgmlv2_d_bound_betamin},
+\ref{eqn:bgmlv3_d_bound_betamin},
+\ref{eqn:positive_rad_d_bound_betamin},
+this happens when:
+
+\begin{equation}
+ \min\left(
+ \sage{bgmlv2_d_upperbound_exp_term},
+ \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)},
+ \right)
+ < \frac{1}{\lcm(m,2n^2)}
+\end{equation}
\minorheading{Irrational $\beta$}
--
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