From 27e90b5f3b84e58577f411d56a1e1582deb62850 Mon Sep 17 00:00:00 2001
From: t0wbo2t <52655804+t0wbo2t@users.noreply.github.com>
Date: Thu, 15 May 2025 10:18:15 +0530
Subject: [PATCH 1/6] Update factorization.md [Update Powersmooth Definition]

The definition of powersmooth was a bit confusing so I added a formal definition inspired by a number theory book.
---
 src/algebra/factorization.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/algebra/factorization.md b/src/algebra/factorization.md
index 11bf4049c..51f206482 100644
--- a/src/algebra/factorization.md
+++ b/src/algebra/factorization.md
@@ -160,7 +160,7 @@ By looking at the squares $a^2$ modulo a fixed small number, it can be observed
 ## Pollard's $p - 1$ method { data-toc-label="Pollard's <script type='math/tex'>p - 1</script> method" }
 
 It is very likely that at least one factor of a number is $B$**-powersmooth** for small $B$.
-$B$-powersmooth means that every prime power $d^k$ that divides $p-1$ is at most $B$.
+$B$-powersmooth means that every prime power $d^k$ that divides $p-1$ is at most $B$. Formally, let $\mathrm{B} \geqslant 1$ and $n \geqslant 1$ with prime factorization $n = \prod {p_i}^{e_i},$ then $n$ is $\mathrm{B}$-powersmooth if, for all $i,$ ${p_i}^{e_i} \leqslant \mathrm{B}$. 
 E.g. the prime factorization of $4817191$ is $1303 \cdot 3697$.
 And the factors are $31$-powersmooth and $16$-powersmooth respectably, because $1303 - 1 = 2 \cdot 3 \cdot 7 \cdot 31$ and $3697 - 1 = 2^4 \cdot 3 \cdot 7 \cdot 11$.
 In 1974 John Pollard invented a method to extracts $B$-powersmooth factors from a composite number.

From 422fb19bdd7c39a56d8fb51caefa419251b37fab Mon Sep 17 00:00:00 2001
From: t0wbo2t <52655804+t0wbo2t@users.noreply.github.com>
Date: Wed, 21 May 2025 17:33:44 +0530
Subject: [PATCH 2/6] Update factorization.md [Update powersmooth definition
 with suggestions]

Commas are now outside $...$ and definiton is for (p - 1) for consistency.
---
 src/algebra/factorization.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/algebra/factorization.md b/src/algebra/factorization.md
index 51f206482..997a11be2 100644
--- a/src/algebra/factorization.md
+++ b/src/algebra/factorization.md
@@ -160,7 +160,7 @@ By looking at the squares $a^2$ modulo a fixed small number, it can be observed
 ## Pollard's $p - 1$ method { data-toc-label="Pollard's <script type='math/tex'>p - 1</script> method" }
 
 It is very likely that at least one factor of a number is $B$**-powersmooth** for small $B$.
-$B$-powersmooth means that every prime power $d^k$ that divides $p-1$ is at most $B$. Formally, let $\mathrm{B} \geqslant 1$ and $n \geqslant 1$ with prime factorization $n = \prod {p_i}^{e_i},$ then $n$ is $\mathrm{B}$-powersmooth if, for all $i,$ ${p_i}^{e_i} \leqslant \mathrm{B}$. 
+$B$-powersmooth means that every prime power $d^k$ that divides $p-1$ is at most $B$. Formally, let $\mathrm{B} \geqslant 1$ and let $p$ be a prime such that $(p - 1) \geqslant 1$. Suppose the prime factorization of $(p - 1)$ is $(p - 1) = \prod {q_i}^{e_i}$, where each $q_i$ is a prime and $e_i \geqslant 1$ then $(p - 1)$ is $\mathrm{B}$-powersmooth if, for all $i$, ${q_i}^{e_i} \leqslant \mathrm{B}$. 
 E.g. the prime factorization of $4817191$ is $1303 \cdot 3697$.
 And the factors are $31$-powersmooth and $16$-powersmooth respectably, because $1303 - 1 = 2 \cdot 3 \cdot 7 \cdot 31$ and $3697 - 1 = 2^4 \cdot 3 \cdot 7 \cdot 11$.
 In 1974 John Pollard invented a method to extracts $B$-powersmooth factors from a composite number.

From 1a7db31f720455b647a929596ff70936875f7532 Mon Sep 17 00:00:00 2001
From: t0wbo2t <52655804+t0wbo2t@users.noreply.github.com>
Date: Thu, 22 May 2025 09:24:40 +0530
Subject: [PATCH 3/6] Update factorization.md [Update Pollard's (p - 1) Method]

Updated materials related to powersmoothness.
Corrected some minor mistakes.
---
 src/algebra/factorization.md | 13 ++++++-------
 1 file changed, 6 insertions(+), 7 deletions(-)

diff --git a/src/algebra/factorization.md b/src/algebra/factorization.md
index 997a11be2..84bdd356d 100644
--- a/src/algebra/factorization.md
+++ b/src/algebra/factorization.md
@@ -159,11 +159,10 @@ By looking at the squares $a^2$ modulo a fixed small number, it can be observed
 
 ## Pollard's $p - 1$ method { data-toc-label="Pollard's <script type='math/tex'>p - 1</script> method" }
 
-It is very likely that at least one factor of a number is $B$**-powersmooth** for small $B$.
-$B$-powersmooth means that every prime power $d^k$ that divides $p-1$ is at most $B$. Formally, let $\mathrm{B} \geqslant 1$ and let $p$ be a prime such that $(p - 1) \geqslant 1$. Suppose the prime factorization of $(p - 1)$ is $(p - 1) = \prod {q_i}^{e_i}$, where each $q_i$ is a prime and $e_i \geqslant 1$ then $(p - 1)$ is $\mathrm{B}$-powersmooth if, for all $i$, ${q_i}^{e_i} \leqslant \mathrm{B}$. 
+It is very likely that a number $n$ has at least one prime factor $p$ such that $p - 1$ is $\mathrm{B}$**-powersmooth** for small $\mathrm{B}$. An integer $m$ is said to be $\mathrm{B}$-powersmooth if every prime power dividing $m$ is at most $\mathrm{B}$. Formally, let $\mathrm{B} \geqslant 1$ and let $m$ be any positive integer. Suppose the prime factorization of $m$ is $m = \prod {q_i}^{e_i}$, where each $q_i$ is a prime and $e_i \geqslant 1$. Then $m$ is $\mathrm{B}$-powersmooth if, for all $i$, ${q_i}^{e_i} \leqslant \mathrm{B}$. 
 E.g. the prime factorization of $4817191$ is $1303 \cdot 3697$.
-And the factors are $31$-powersmooth and $16$-powersmooth respectably, because $1303 - 1 = 2 \cdot 3 \cdot 7 \cdot 31$ and $3697 - 1 = 2^4 \cdot 3 \cdot 7 \cdot 11$.
-In 1974 John Pollard invented a method to extracts $B$-powersmooth factors from a composite number.
+And the values, $1303 - 1$ and $3697 - 1$, are $31$-powersmooth and $16$-powersmooth respectively, because $1303 - 1 = 2 \cdot 3 \cdot 7 \cdot 31$ and $3697 - 1 = 2^4 \cdot 3 \cdot 7 \cdot 11$.
+In 1974 John Pollard invented a method to extracts $\mathrm{B}$-powersmooth factors from a composite number.
 
 The idea comes from [Fermat's little theorem](phi-function.md#application).
 Let a factorization of $n$ be $n = p \cdot q$.
@@ -180,7 +179,7 @@ This means that $a^M - 1 = p \cdot r$, and because of that also $p ~|~ \gcd(a^M
 
 Therefore, if $p - 1$ for a factor $p$ of $n$ divides $M$, we can extract a factor using [Euclid's algorithm](euclid-algorithm.md).
 
-It is clear, that the smallest $M$ that is a multiple of every $B$-powersmooth number is $\text{lcm}(1,~2~,3~,4~,~\dots,~B)$.
+It is clear, that the smallest $M$ that is a multiple of every $\mathrm{B}$-powersmooth number is $\text{lcm}(1,~2~,3~,4~,~\dots,~B)$.
 Or alternatively:
 
 $$M = \prod_{\text{prime } q \le B} q^{\lfloor \log_q B \rfloor}$$
@@ -189,11 +188,11 @@ Notice, if $p-1$ divides $M$ for all prime factors $p$ of $n$, then $\gcd(a^M -
 In this case we don't receive a factor.
 Therefore, we will try to perform the $\gcd$ multiple times, while we compute $M$.
 
-Some composite numbers don't have $B$-powersmooth factors for small $B$.
+Some composite numbers don't have $\mathrm{B}$-powersmooth factors for small $\mathrm{B}$.
 For example, the factors of the composite number $100~000~000~000~000~493 = 763~013 \cdot 131~059~365~961$ are $190~753$-powersmooth and $1~092~161~383$-powersmooth.
 We will have to choose $B >= 190~753$ to factorize the number.
 
-In the following implementation we start with $B = 10$ and increase $B$ after each each iteration.
+In the following implementation we start with $\mathrm{B} = 10$ and increase $\mathrm{B}$ after each each iteration.
 
 ```{.cpp file=factorization_p_minus_1}
 long long pollards_p_minus_1(long long n) {

From 2597e0558304678a7fa7f92c66a19f9f7913c974 Mon Sep 17 00:00:00 2001
From: t0wbo2t <52655804+t0wbo2t@users.noreply.github.com>
Date: Thu, 29 May 2025 11:19:43 +0530
Subject: [PATCH 4/6] Update src/algebra/factorization.md

Co-authored-by: Oleksandr Kulkov <adamant.pwn@gmail.com>
---
 src/algebra/factorization.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/algebra/factorization.md b/src/algebra/factorization.md
index 84bdd356d..bc606607f 100644
--- a/src/algebra/factorization.md
+++ b/src/algebra/factorization.md
@@ -188,8 +188,8 @@ Notice, if $p-1$ divides $M$ for all prime factors $p$ of $n$, then $\gcd(a^M -
 In this case we don't receive a factor.
 Therefore, we will try to perform the $\gcd$ multiple times, while we compute $M$.
 
-Some composite numbers don't have $\mathrm{B}$-powersmooth factors for small $\mathrm{B}$.
-For example, the factors of the composite number $100~000~000~000~000~493 = 763~013 \cdot 131~059~365~961$ are $190~753$-powersmooth and $1~092~161~383$-powersmooth.
+Some composite numbers don't have factors $p$ s.t. $p-1$ is $\mathrm{B}$-powersmooth for small $\mathrm{B}$.
+For example, for the composite number $100~000~000~000~000~493 = 763~013 \cdot 131~059~365~961$, values $p-1$ are $190~753$-powersmooth and $1~092~161~383$-powersmooth correspondingly.
 We will have to choose $B >= 190~753$ to factorize the number.
 
 In the following implementation we start with $\mathrm{B} = 10$ and increase $\mathrm{B}$ after each each iteration.

From 54fec62526c6454ecd43b38544c6a56880031544 Mon Sep 17 00:00:00 2001
From: t0wbo2t <52655804+t0wbo2t@users.noreply.github.com>
Date: Thu, 29 May 2025 11:19:54 +0530
Subject: [PATCH 5/6] Update src/algebra/factorization.md

Co-authored-by: Oleksandr Kulkov <adamant.pwn@gmail.com>
---
 src/algebra/factorization.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/algebra/factorization.md b/src/algebra/factorization.md
index bc606607f..9d9ab7ed7 100644
--- a/src/algebra/factorization.md
+++ b/src/algebra/factorization.md
@@ -162,7 +162,7 @@ By looking at the squares $a^2$ modulo a fixed small number, it can be observed
 It is very likely that a number $n$ has at least one prime factor $p$ such that $p - 1$ is $\mathrm{B}$**-powersmooth** for small $\mathrm{B}$. An integer $m$ is said to be $\mathrm{B}$-powersmooth if every prime power dividing $m$ is at most $\mathrm{B}$. Formally, let $\mathrm{B} \geqslant 1$ and let $m$ be any positive integer. Suppose the prime factorization of $m$ is $m = \prod {q_i}^{e_i}$, where each $q_i$ is a prime and $e_i \geqslant 1$. Then $m$ is $\mathrm{B}$-powersmooth if, for all $i$, ${q_i}^{e_i} \leqslant \mathrm{B}$. 
 E.g. the prime factorization of $4817191$ is $1303 \cdot 3697$.
 And the values, $1303 - 1$ and $3697 - 1$, are $31$-powersmooth and $16$-powersmooth respectively, because $1303 - 1 = 2 \cdot 3 \cdot 7 \cdot 31$ and $3697 - 1 = 2^4 \cdot 3 \cdot 7 \cdot 11$.
-In 1974 John Pollard invented a method to extracts $\mathrm{B}$-powersmooth factors from a composite number.
+In 1974 John Pollard invented a method to extract factors $p$, s.t. $p-1$ is $\mathrm{B}$-powersmooth, from a composite number.
 
 The idea comes from [Fermat's little theorem](phi-function.md#application).
 Let a factorization of $n$ be $n = p \cdot q$.

From 53639d500284ae160ca2e9f968957c360b6f2040 Mon Sep 17 00:00:00 2001
From: t0wbo2t <52655804+t0wbo2t@users.noreply.github.com>
Date: Thu, 29 May 2025 11:20:00 +0530
Subject: [PATCH 6/6] Update src/algebra/factorization.md

Co-authored-by: Oleksandr Kulkov <adamant.pwn@gmail.com>
---
 src/algebra/factorization.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/algebra/factorization.md b/src/algebra/factorization.md
index 9d9ab7ed7..14715605f 100644
--- a/src/algebra/factorization.md
+++ b/src/algebra/factorization.md
@@ -190,7 +190,7 @@ Therefore, we will try to perform the $\gcd$ multiple times, while we compute $M
 
 Some composite numbers don't have factors $p$ s.t. $p-1$ is $\mathrm{B}$-powersmooth for small $\mathrm{B}$.
 For example, for the composite number $100~000~000~000~000~493 = 763~013 \cdot 131~059~365~961$, values $p-1$ are $190~753$-powersmooth and $1~092~161~383$-powersmooth correspondingly.
-We will have to choose $B >= 190~753$ to factorize the number.
+We will have to choose $B \geq 190~753$ to factorize the number.
 
 In the following implementation we start with $\mathrm{B} = 10$ and increase $\mathrm{B}$ after each each iteration.
 

<!DOCTYPE html PUBLIC '-//W3C//DTD XHTML 1.0 Transitional//EN' 'http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd'>
<html xmlns='http://www.w3.org/1999/xhtml'>
<head>
<title>pFad - Phonifier reborn</title>
<meta http-equiv='Content-Type' content='text/html; charset=utf-8' />
</head>
<body>
<h1>Pfad - The Proxy pFad of &#169; 2024 Garber Painting. All rights reserved.</h1>


<!-- Disclaimer -->
<p>Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.</p>
<br>
<p>Alternative Proxies:</p><p><a href="http://rainy.clevelandohioweatherforecast.com/php-proxy/index.php?q=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1461.patch" target="_blank">Alternative Proxy</a></p><p><a href="http://rainy.clevelandohioweatherforecast.com/pFad/index.php?u=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1461.patch" target="_blank">pFad Proxy</a></p><p><a href="http://rainy.clevelandohioweatherforecast.com/pFad/v3index.php?u=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1461.patch" target="_blank">pFad v3 Proxy</a></p><p><a href="http://rainy.clevelandohioweatherforecast.com/pFad/v4index.php?u=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1461.patch" target="_blank">pFad v4 Proxy</a></p></body>
</html>