# Is the number of ALL primes even or odd?

### 10 Answers

- RaymondLv 78 months ago
It was already known over 2000 years ago (by Greek geometers) that there is no such things as a highest prime number. It was later proved that the number of primes is infinite.

The proof goes something like this (it is a proof by contradiction).

Suppose that the number of primes is really finite.

Then, there must exist a list of them, and it would go like this:

2, 3, 5, 7, 11, 13, ... x, y, z

It does not matter what x, y and z are. The only thing we need is that: "if the list is finite, then they have a definite value".

We can create a number by multiplying all the primes together:

N = 2 * 3 * 4 * 5 * 7 * 11 * 13 * ... * x * y * z

Obviously, N is not a prime number. However, it is obviously greater than z, our supposed "greatest prime number". And, just as obviously, it is divisible by 2, 3, 5, 7... and all the prime numbers up to z. It is, for example, an even number, divisible by 3, by 5 and so on.

What about N+1 ?

It has to be an odd number (even + 1 = odd) therefore it is not divisible by 2.

If you divide it by 3, it will leave a remainder of 1 (therefore, N+1 is not divisible by 3)

and so on for all known primes.

N+1 is not divisible by any of the prime numbers up to z (our last prime number), therefore N+1 must also be prime - and it is greater than z.

The idea that the list of primes is finite, leads to a contradiction. Therefore, the idea is false. The list of prime numbers is infinite. Infinity is not a well-defined number: it is neither even nor odd (in number theory, infinity is NOT a "number")

- ToddLv 78 months ago
There is no ceiling for the number of prime numbers. The only thing that can be proven is that the number of primes grows but decelerates rapidly. That's why the sieve of Eratosthones is so fast.

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- 8 months ago
The sum of all primes is infinity and as a concept, that is neither even nor odd.

- LônLv 78 months ago
Impossible to tell

2 + 3 = 5....odd

2 + 3 + 5 = 10.. even

2 + 3 + 5 + 7 = 17 ...odd

This keeps repeating.... where do you stop??