Prime Numbers

Question 1

What is Lemma 3.1?

Answer 1

Let a, b be integers
1. If p is irreducible, either p divides a or a and p are coprime
2. If p is prime then it is irreducible
3. If p is irreducible then it is also prime

Question 2

What does irreducible mean?

Answer 2

It means that the only divisors of p are 1 and p

Question 3

What does prime mean?

Answer 3

P is said to be prime if for all integers a and b, if p|ab then
p|a or p|b

Question 4

What is an integer that is not irreducible called?

Answer 4

Composite and it has the form n=ab

Question 5

What is corollary 3.2?

Answer 5

If p is prime and divides a1…ak, then p divides ai for some I.

Question 6

What is theorem 3.3 (The Fundamental theorem of Arithmetic)?

Answer 6

Each integer n>1 has a prime-power factorisation
n=p1^e1…pk^ek
Where p1…pk are distinct primes and e1…ek are positive integers, this factorisation is unique, apart from permutations of the factors

Question 7

What is corollary 3.6?

Answer 7

If a positive integer m is not a perfect square then root(m) is irrational

Question 8

What is theorem 3.7?

Answer 8

There are infinitely many primes

Question 9

What is corollary 3.9?

Answer 9

The n-th prime pn satisfies pn <= 2^(2n-1) for all n>= 1