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# Applications of prime factorisation

Lemma 6.6.1   If is not a square number then is irrational.

Proof. Suppose , with . Then . If then let be a prime dividing . Thus and so , which is impossible as . Thus and . This lemma can also be stated: if with then ''.

Definition 6.6.2   A real number is algebraic if it satisfies a polynomial equation with coefficients in . Real numbers which are not algebraic are transcendental

For instance and are transcendental (the proof is fairly hard, especially for ). 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 ( with ) satisfies a polynomial with coefficients in then so and . In particular if then , which is stated as algebraic integers which are rational are integers''. Note that if and with then and , and . 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 and '').    Next: Modular Arithmetic Up: The classical sets of Previous: Uniqueness of prime factorisation   Contents
root 2002-06-10