How do you prove there's an infinite number of prime numbers?

Answer 1

You make the assumption that there are a finite number of primes.
If this is so, then it is possible to write them all out in a list.
You then multiply them all together, and add one to the product.
The result is not divisible by any of your existing prime numbers. So it is either the case that:
It is not divisible by any number, i.e. it is prime.
It is divisible by two or more prime numbers, which cannot be on your list.
Either way, you have missed out at least one prime number. Therefore, not matter what your list of prime numbers, you have always missed out some, which means that there is an infinite number of primes.

Answer 2

The proof depends on the fact that every natural number has a unique factorisation into primes. Now assume that there is only a finite number of primes and let's call these p_1, p_2, ... , p_m.(p_m is largest) let (p_1)*(p_2)*...*(p_m) = q Look at the number q + 1 The above is either prime or composite. Now if it is a prime we have reached a contradiction. if it is composite then the factor > p_m . becuase no number(other than 1) can be factor of q and q+ 1 beacuse in this case it has to be factor of 1 again a contradiction so number of prime numbers infinite.

Answer 3

the technique used to prove that an infinite number of prime numbers exist is called "proof by contradiction".

start with the assumption that the number of prime numbers is finite. you can show that this assumption leads to a contradiction.
this is proof that the assumption (the number of prime numbers is finite) is false. this proves that the number of primes is infinite.

Answer 4

The proof depends on the fact that every natural number has a unique prime factorisation

Now assume that there is only a finite number of primes and let's call
these p_1, p_2, ... , p_m.

Look at the number p = (p_1)*(p_2)*...*(p_m) + 1

There are 2 possibilities

it it is a prime we have reached a contradiction.

If it is not a prime then prime factor > p_m.

This is because if p_k is a factor of p p_k <= p_m
so p_k is a factor or 1 becuase p_k is factor of p+1

so prime factor > p_m

so p_m is not larget prime

so contradiction in both cases.

So infinite prime numbers

Answer 5

Because there are an infinite number of numbers and prime numbers will always happen where numbers happen so there will be an infinite number of prime numbers.

Answer 6

The proofs given are variations of Euclid's original proof that appears in Elements, it is one that employs contradiction.

consider the following proof due to Euler,
http://en.wikipedia.org/wiki/Proof_that_the_sum_of_the_reciprocals_of_the_primes_diverges