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Applications of prime factorisation

Lemma 6.6.1   If $ n \in \mathbb{N}$ is not a square number then $ \sqrt{n}$ is irrational.

Proof. Suppose $ \sqrt{n} = \frac{a}{b}$, with $ (a,b) = 1$. Then $ n b^2 = a^2$. If $ b > 1$ then let $ p$ be a prime dividing $ b$. Thus $ p \mid a^2$ and so $ p \mid a$, which is impossible as $ (a,b) = 1$. Thus $ b = 1$ and $ n = a^2$. $ \qedsymbol$

This lemma can also be stated: ``if $ n \in N$ with $ \sqrt{n} \in \mathbb{Q}$ then $ \sqrt{n} \in \mathbb{N}$''.

Definition 6.6.2   A real number is algebraic if it satisfies a polynomial equation with coefficients in $ \mathbb{Z}$. Real numbers which are not algebraic are transcendental

For instance $ \pi$ and $ e$ are transcendental (the proof is fairly hard, especially for $ \pi$). Most reals are transcendental; this can be proven using the notion of a countable set and we will do it later (v.i. Theorem thm transcendental numbers -> uncountable). If the rational $ \frac{a}{b}$ ( with $ (a,b) = 1$ ) satisfies a polynomial with coefficients in $ \mathbb{Z}$ then

$\displaystyle c_n a^n + c_{n-1} a^{n-1} b + \dots b^n c_0 = 0
$

so $ b \mid c_n$ and $ a \mid c_0$. In particular if $ c_n = 1$ then $ b = 1$, which is stated as ``algebraic integers which are rational are integers''. Note that if $ a = p_1^{\alpha_1} p_2^{\alpha_2} \dots p_k^{\alpha_k}$ and $ b = p_1^{\beta_1} p_2^{\beta_2} \dots p_k^{\beta_k}$ with $ \alpha_i, \beta_i \in \mathbb{N}_0$ then $ (a,b) = p_1^{\gamma_1}
p_2^{\gamma_2} \dots p_k^{\gamma_k}$ and $ [a,b] = p_1^{\delta_1}
p_2^{\delta_2} \dots p_k^{\delta_k}$, $ \gamma_i = \min \{\alpha_i,
\beta_i\}$ and $ \delta_i = \max \{ \alpha_i, \beta_i \}$. Major open problems in the area of prime numbers are the Goldbach conjecture (``every even number greater than two is the sum of two primes'') and the twin primes conjecture (``there are infinitely many prime pairs $ p$ and $ p+2$'').
next up previous contents
Next: Modular Arithmetic Up: The classical sets of Previous: Uniqueness of prime factorisation   Contents
root 2002-06-10