Number Theory

Core Theoretical Principles

Number theory explores the discrete structure of the integers $\mathbb{Z}$, prime factorization, and modular arithmetic.

1. Euclid’s Formula for Primitive Pythagorean Triples

A Pythagorean triple consists of three natural numbers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$. All primitive triples (where $\gcd(a, b, c) = 1$) can be generated by choosing coprime integers $m > n > 0$ with opposite parity:

$$ a = m^2 - n^2, \qquad b = 2mn, \qquad c = m^2 + n^2 $$

Connecting this to the unit circle $x^2 + y^2 = 1$ via rational parametrization $(x, y) = \left(\frac{m^2-n^2}{m^2+n^2}, \frac{2mn}{m^2+n^2}\right)$ gives an intuitive geometric link between trigonometry and number theory.

2. Euler’s Totient Function $\phi(n)$

Euler’s totient function counts the number of positive integers up to $n$ that are relatively prime to $n$:

$$ \phi(n) = n \prod_{p \mid n} \left(1 - \frac{1}{p}\right) $$

This function plays a fundamental role in counting Pythagorean triples and in modern RSA public-key cryptography via Euler’s Totient Theorem:

$$ a^{\phi(n)} \equiv 1 \pmod{n} \quad \text{for } \gcd(a, n) = 1 $$

3. Irrationality of $\sqrt{3}$

We prove that $\sqrt{3} \notin \mathbb{Q}$ using proof by contradiction. If $\sqrt{3} = \frac{p}{q}$ with $\gcd(p, q) = 1$, then:

$$ 3q^2 = p^2 \implies 3 \mid p \implies p = 3k \implies 3q^2 = 9k^2 \implies q^2 = 3k^2 \implies 3 \mid q $$

This contradicts the assumption that $\gcd(p, q) = 1$.