Prove-that-P-set-of-prime-numbers-is-countable-

Question Number 3551 by prakash jain last updated on 15/Dec/15

Prove that P set of prime numbers is

countable .

Commented by Filup last updated on 15/Dec/15

ah, i understand the question now .

I {can}^{'} t answer it, though

Commented by Filup last updated on 15/Dec/15

If by countable, you mean there

is a finite number of primes, t h e n y o u

a r e i n c o r r e c t .

Reason is :

a l l n u m b e r s h a v e p r i m e f a c t o r s

let n \in Z

∴ n = p_{1} \times \dots \times p_{k} p r i m e f a c t o r s

p_{i} \in P

n + 1 \neq n ∴ d i f f e r e n t prime factors .

all primes are prime factors of a number

beacuse p_{1} \times \dots \times p_{k} = n

a single prime is the prime factor of an

infinite amount of numbers

Commented by prakash jain last updated on 15/Dec/15

A countable set may be finite or infinite .

Essentially it means that every element

of set can be associated with natural number .

Commented by prakash jain last updated on 15/Dec/15

P \subset N, since N is countable P is countable .