From 90aef21977b04fef7c6e1fa371df9a796d22d437 Mon Sep 17 00:00:00 2001 From: TimShatnyy <64311787+TimShatnyy@users.noreply.github.com> Date: Tue, 4 Nov 2025 21:06:05 +0100 Subject: [PATCH 1/2] Motivation for learning Number theory --- pages/content/ch4.mdx | 29 ++++++++++++++++++++++++ pages/content/ch4/division.mdx | 1 + pages/content/ch4/modular-arithmetic.mdx | 3 +++ 3 files changed, 33 insertions(+) diff --git a/pages/content/ch4.mdx b/pages/content/ch4.mdx index 1431e96..154449e 100644 --- a/pages/content/ch4.mdx +++ b/pages/content/ch4.mdx @@ -1,6 +1,33 @@ +import { Callout, Tabs } from 'nextra/components' + # Number Theory +The Number Theory is a wonderful part of mathematics which studies how to solve various problems using properties that different numbers possess. + +## Motivation + +As a **motivation for studying**, the following example might be good: + + +**Is there a solution of $x^3 + x^2 = y^4 + y + 1$?** + +Aaaand ... No there is no solution due to the fact that both sides can't be at the same time $odd = odd$ or $even = even$. + +**Let us take two cases:** +1. **y = even**, then when we have $y^4 + y + 1$, then $y^4$ is as well even,. Now, adding $(y^4) \ even + (y) \ even = even$. Finally, $((y^4 + y) + 1)$ results in $even + odd$, which is always $odd$. +2. **y = odd**, then $y^4 + y + 1 = odd + odd + odd = odd$ + +So the LHS is always odd. Now, considering LHS, we can do a **similar analysis**: + +1. **x - even**, then $x^3 + x^2 = even + even = even$ +2. **x - odd**, then $x^3 + x^2 = odd + odd = even$ + +RHS is always $even$. Here we arrive at a contradiction ⚡️ +$even \neq odd$, which proves, that there is no solution to the given equation. + + +This **example reveals** that by knowing how even and odd numbers react to adding them or raising to some power (in a sense **knowing some theory behind these numbers**), we can prove that there is no solution for a given equation. ## References @@ -12,3 +39,5 @@ The following are some references to problem sets which were quite useful to get - [*Number Theory Document* - artofproblemsolving.com](https://artofproblemsolving.com/articles/files/SatoNT.pdf?srsltid=AfmBOopyi3E0r3pHpuDR7Oi9LV55TyzI33f5T_rF-udQyytDC7x8g6za) + + diff --git a/pages/content/ch4/division.mdx b/pages/content/ch4/division.mdx index 86e219d..e26faf8 100644 --- a/pages/content/ch4/division.mdx +++ b/pages/content/ch4/division.mdx @@ -19,6 +19,7 @@ import { Cards } from 'nextra/components' /> + **Definition 4.1**:
For integers $a$ and $b$, we say that **$a$ divides $b$**, denoted $a \mid b$, if there exists an integer $c$ such that $b = ac$. In this case, $a$ is called a **divisor** of $b$, and $b$ is called a **multiple** of $a$. If $a \neq 0$ and a divisor $c$ exists, it is called the **quotient** when $b$ is divided by $a$, and we write $$c = \frac{b}{a}$$ or $c = b/a$. 4. We write $a \nmid b$ if $a$ does not divide $b$. diff --git a/pages/content/ch4/modular-arithmetic.mdx b/pages/content/ch4/modular-arithmetic.mdx index ddef6d4..8e4cf94 100644 --- a/pages/content/ch4/modular-arithmetic.mdx +++ b/pages/content/ch4/modular-arithmetic.mdx @@ -7,3 +7,6 @@ import { Cards } from 'nextra/components' # Modular Arihmetic + + + From 8a1952ea6c58dbf00cc86939d18223553a4a4a6b Mon Sep 17 00:00:00 2001 From: TimShatnyy <64311787+TimShatnyy@users.noreply.github.com> Date: Wed, 5 Nov 2025 13:26:03 +0100 Subject: [PATCH 2/2] Corrections on motivation for Number Theory motivation. --- pages/content/ch4.mdx | 20 ++++++++++++-------- 1 file changed, 12 insertions(+), 8 deletions(-) diff --git a/pages/content/ch4.mdx b/pages/content/ch4.mdx index 154449e..1aa56b9 100644 --- a/pages/content/ch4.mdx +++ b/pages/content/ch4.mdx @@ -10,21 +10,25 @@ The Number Theory is a wonderful part of mathematics which studies how to solve As a **motivation for studying**, the following example might be good: -**Is there a solution of $x^3 + x^2 = y^4 + y + 1$?** +**Is there a solution of $x^3 + x^2 = y^4 + y + 1$ for some integers $x$, $y$?** -Aaaand ... No there is no solution due to the fact that both sides can't be at the same time $odd = odd$ or $even = even$. +Aaaand ... No there is no solution due to the fact that both sides can't be at the same time: + +$$ +\text{odd} = \text{odd} \text{ or } \text{even} = \text{even} +$$ **Let us take two cases:** -1. **y = even**, then when we have $y^4 + y + 1$, then $y^4$ is as well even,. Now, adding $(y^4) \ even + (y) \ even = even$. Finally, $((y^4 + y) + 1)$ results in $even + odd$, which is always $odd$. -2. **y = odd**, then $y^4 + y + 1 = odd + odd + odd = odd$ +1. **$y = \text{even}$**, then when we have $y^4 + y + 1$, then $y^4$ is as well even,. Now, adding $(y^4) \ \text{even} + (y) \ \text{even} = \text{even}$. Finally, $((y^4 + y) + 1)$ results in $\text{even} + \text{odd}$, which is always $\text{odd}$. +2. **$y = \text{odd}$**, then $y^4 + y + 1 = \text{odd} + \text{odd} + \text{odd} = \text{odd}$ So the LHS is always odd. Now, considering LHS, we can do a **similar analysis**: -1. **x - even**, then $x^3 + x^2 = even + even = even$ -2. **x - odd**, then $x^3 + x^2 = odd + odd = even$ +1. **$x = \text{even}$**, then $x^3 + x^2 = \text{even} + \text{even} = \text{even}$ +2. **$x = \text{odd}$**, then $x^3 + x^2 = \text{odd} + \text{odd} = \text{even}$ -RHS is always $even$. Here we arrive at a contradiction ⚡️ -$even \neq odd$, which proves, that there is no solution to the given equation. +RHS is always $\text{even}$. Here we arrive at a **contradiction** ⚡️ +$\text{even} \neq \text{odd}$, which proves, that there is no solution to the given equation. This **example reveals** that by knowing how even and odd numbers react to adding them or raising to some power (in a sense **knowing some theory behind these numbers**), we can prove that there is no solution for a given equation.