From d43302bc842352ed6794dd3abbab8c1cfe265c26 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 30 May 2023 15:08:08 +0100
Subject: [PATCH] Change superscripts to subscripts on the k
---
main.tex | 12 ++++++------
1 file changed, 6 insertions(+), 6 deletions(-)
diff --git a/main.tex b/main.tex
index d3020c5..d2b3567 100644
--- a/main.tex
+++ b/main.tex
@@ -1112,18 +1112,18 @@ Where $\epsilon_{q,1}$ and $\epsilon_{q,2}$ are defined as follows:
\begin{equation*}
\epsilon_{q,1} :=
- \frac{k_q^1}{2mn^2}
+ \frac{k_{q,1}}{2mn^2}
\qquad
\epsilon_{q,2} :=
- \frac{k_q^2}{2mn^2}
+ \frac{k_{q,2}}{2mn^2}
\end{equation*}
\begin{align*}
\text{where }
- &k_q^1 \text{ is the least }
+ &k_{q,1} \text{ is the least }
k\in\ZZ_{>0}\: s.t.:\:
k \equiv -\aa\bb m \mod n
\\
- &k_q^2 \text{ is the least }
+ &k_{q,2} \text{ is the least }
k\in\ZZ_{>0}\: s.t.:\:
k \equiv \aa\bb m (\aa\aa^{'}-2)
\mod n\gcd(2n,\aa^2 m)
@@ -1174,13 +1174,13 @@ Consider the following tautology:
In our situation, we want to find the least $k$ satisfying
eqn \ref{eqn:finding_better_eps_problem}.
Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
-we can pick the smallest $k_q^1 \in \ZZ_{>0}$ which satisfies this new condition
+we can pick the smallest $k_{q,1} \in \ZZ_{>0}$ which satisfies this new condition
(a computation only depending on $q$ and $\beta$, but not $r$).
We are then guaranteed that the gap $\frac{k}{2mn^2}$ is at least
$\epsilon_{q,1}$.
Furthermore, $k$ also satisfies
eqn \ref{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
-so we can also pick the smallest $k_q^2 \in \ZZ_{>0}$ satisfying this condition,
+so we can also pick the smallest $k_{q,2} \in \ZZ_{>0}$ satisfying this condition,
which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_{q,2}$.
\end{proof}
--
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