Question

How to prove that there are infinite prime numbers?

Answer 1

Let's assume to reach our contradiction that there are only finite prime numbers. Thus, let the largest and the final prime number be p. So our assumption means that there are only finite prime numbers and those are from 2 to p.

Now, multiply the known prime numbers.

2 × 3 × 5 × 7 ×......× p

Then add 1 to this product.

(2 × 3 × 5 × 7 ×......× p) + 1

This number obtained should be greater than p. But when we see this number deeper, we get that this number leaves remainder 1 on division by the known and only finite prime numbers from 2 to p.

From this we get two facts that either this number is a prime number itself or this number can be divided by another prime number greater than p.

Even any of these two, these contradict our earlier assumption that p is not the largest prime number.

Hence proved!!!

Answer 2

Let there be definite prime intergers like p₁  p₂ ....... pₙ. Assuming their product to be x.

x = p₁ p₂ ...... pₙ

Here x was divisible by each prime mentioned.

On adding 1 both sides we get,

x + 1 = 1 + p₁ p₂ ..... pₙ

x' = 1 + p₁ p₂ ..... pₙ

Undoubtedly, x' is no more divisible by any of the prime and this contradicts our assumption that there are definite prime numbers.

There must be some other which could divide x'. Hence there are infinte prime integers.

Q.E.D