Merge pull request #8 from kolosovpetro/MATH-100

MATH-100. Minimal Goldbach pairs in prime counting

Author: kolosovpetro

CHANGELOG.md

@@ -5,6 +5,14 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec/v2.0.0.html).
 
+## [1.1.1] - 2025-04-22
+
+### Changed
+
+- Minor edits in text
+- Add Acknowledgements
+- Grammar fixes
+
 ## [1.1.0] - 2025-04-22
 
 ### Changed

Folder.DotSettings

@@ -0,0 +1,3 @@
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+	<s:Boolean x:Key="/Default/UserDictionary/Words/=beleza/@EntryIndexedValue">True</s:Boolean>
+	<s:Boolean x:Key="/Default/UserDictionary/Words/=yamagishi/@EntryIndexedValue">True</s:Boolean></wpf:ResourceDictionary>

out/MinimalGoldbachPairsInPrimesCounting.pdf

@@ -75,15 +75,25 @@
 
     \maketitle
 
-%    \tableofcontents
+    \tableofcontents
 
     \freefootnote{Sources: \url{https://github.com/kolosovpetro/MinimalGoldbachPairsInPrimesCounting}}
 
+
     \section{Introduction} \label{sec:introduction}
     \input{sections/02_introduction}
 
+
+%    \section{Discussion on Twin Primes}\label{sec:discussion-on-twin-primes}
+%    \input{sections/03_discussion_on_twin_primes}
+
+
     \section{Conclusions}\label{sec:conclusions}
-    \input{sections/03_conclusions}
+    \input{sections/conclusions}
+
+
+    \section{Acknowledgements}\label{sec:acknowledgements}
+    \input{sections/acknowledgements}
 
 
     \bibliographystyle{unsrt}

src/MinimalGoldbachPairsInPrimesCounting.tex

@@ -14,22 +14,17 @@
 For the even integer $10$, we have three pairs: $(3,7), \; (5,5), \; (7,3)$
 and the minimal one is $(3,7)$ because
 $3$ is the smallest among all $p_i$ values: $3, 5, 7$.
-
-
 Consider the following minimal Goldbach pairs for even integer $2k$ within the range $6 \leq 2k \leq 50$
 \begin{align*}
-    6  &= 3 + 3,   & 30 &= 7 + 23, \\
-    8  &= 3 + 5,   & 32 &= 3 + 29, \\
-    10 &= 3 + 7,   & 34 &= 3 + 31, \\
-    12 &= 5 + 7,   & 36 &= 5 + 31, \\
-    14 &= 3 + 11,  & 38 &= 7 + 31, \\
-    16 &= 3 + 13,  & 40 &= 3 + 37, \\
-    18 &= 5 + 13,  & 42 &= 5 + 37, \\
-    20 &= 3 + 17,  & 44 &= 3 + 41, \\
-    22 &= 3 + 19,  & 46 &= 3 + 43, \\
-    24 &= 5 + 19,  & 48 &= 5 + 43, \\
-    26 &= 3 + 23,  & 50 &= 3 + 47, \\
-    28 &= 5 + 23,  &
+    6  &= 3 + 3,   & 24 &= 5 + 19,  & 40 &= 3 + 37, \\
+    8  &= 3 + 5,   & 26 &= 3 + 23,  & 42 &= 5 + 37, \\
+    10 &= 3 + 7,   & 28 &= 5 + 23,  & 44 &= 3 + 41, \\
+    12 &= 5 + 7,   & 30 &= 7 + 23,  & 46 &= 3 + 43, \\
+    14 &= 3 + 11,  & 32 &= 3 + 29,  & 48 &= 5 + 43, \\
+    16 &= 3 + 13,  & 34 &= 3 + 31,  & 50 &= 3 + 47, \\
+    18 &= 5 + 13,  & 36 &= 5 + 31,  &             \\
+    20 &= 3 + 17,  & 38 &= 7 + 31,  &             \\
+    22 &= 3 + 19,  &                &
 \end{align*}
 We can notice that minimal Goldbach pairs with minimal $p_i=3$ produce a sequence of odd prime numbers
 $p_j = 3, \; 5, \; 7, \; 11, \; 13, \; 17 \; \dots$ which is quite remarkable:
@@ -55,9 +50,8 @@
 Let $G_{\min} (2N)$ be a function that returns a set of minimal Goldbach pairs $(p_i, p_j)$ having $\min p_i$
 over the range $6 \leq 2k \leq 2N$
 \begin{align*}
-    G_{\min} (2N) = \{(p_i, p_j) \mid p_i + p_j = 2k \mid 6 \leq 2k \leq 2N \mid \min p_i \}.
+    G_{\min} (2N) = \{(p_i, p_j) \mid p_i + p_j = 2k \mid 6 \leq 2k \leq 2N \mid \min p_i \}
 \end{align*}
-
 For example,
 \begin{align*}
     G_{\min}(20) = \{
@@ -94,9 +88,16 @@
 Hence, by excluding the elements of the set $W_{2N}(5)$ from the set $W_{2N}(3)$ yields the sequence of
 lesser twin primes~\cite{oeis:A001359}
 \begin{align*}
-    W_{2N}(3) \setminus W_{2N}(5) = \{ 3, \; 5, \; 11, \; 17, \; 29, \; 41, \; 59, \; 71, \; 101, \; 107, \; 137, \; \dots, \;  p \leq 2N - 3 \}
+    \mathtt{Lesser \; twin \; primes}  &= \{ 3, \; 5, \; 11, \; 17, \; 29, \; 41, \; 59, \; 71, \; 101, \; 107, \; 137, \; \dots, \;  p \leq 2N - 3 \} \\
+    &=  W_{2N}(3) \setminus W_{2N}(5)
 \end{align*}
-This implies that the number of twin primes in range $6 \leq 2k \leq 2N$ can be expressed in terms of $F_{2N}(P)$
+Above relation holds because the function $W_{2N}(5)$ generates the primes $p=6n+1$ such that either
+\begin{itemize}
+    \item $p+2$ is composite: $p+2 = 3(2n+1)$
+    \item $p-2$ is composite or lesser twin prime
+\end{itemize}
+As a result, by excluding $W_{2N}(3) \setminus W_{2N}(5)$ we get the sequence of lesser twin primes.
+This implies that the number of twin primes $\pi_2$ in range $6 \leq 2k \leq 2N$ can be expressed in terms of $F_{2N}(P)$
 \begin{align*}
     \pi_2 (2N) = F_{2N+3}(3) - F_{2N+5}(5)
 \end{align*}

src/sections/02_introduction.tex

@@ -0,0 +1,2 @@
+I am grateful to Michel Eduardo Beleza Yamagishi for incredible insights and ideas related to minimal Goldbach pairs
+and their distribution.