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Number Theory Level 1

$2,3,5,7,11,\ldots$

The above shows an infinite set of prime numbers . How many of these numbers are even numbers ?

1 2 3 4

3 solutions

A Former Brilliant Member
Apr 25, 2016

Relevant wiki: Prime Numbers

$2$ has only two factors $1$ and $2$ . So it is prime.

Every other even number $n$ will be of the form $2k$ . So it must have the factors $1,2,k,n$ so it won't be prime.(for $4$ , the factors are $1,2,4$ and all other even numbers have at least $4$ factors).

Got a proof?

Pi Han Goh - 5 years, 1 month ago

Yes. I have edited my solution. ;)

A Former Brilliant Member - 5 years, 1 month ago

Sorry. That isn't a proof. You just regurgitate the definition of a prime number. You said 2 is the only prime number. To prove this claim, you need to show that all the other (positive) even numbers can't be prime.

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – (added to solution)

A Former Brilliant Member - 5 years, 1 month ago

@A Former Brilliant Member – Great! That's the correct explanation! +1

Can you add that explanation to your solution?

Pi Han Goh - 5 years, 1 month ago

Munem Shahriar
Nov 26, 2017

Statement: Except 2, all prime numbers are odd.

Proof: Even numbers are always divisible by 2. That means except 2, all even numbers have more than two divisors. Hence except 2, all prime numbers must be odd.

Statement: 2 is a prime number.

Proof: 2 is only divisible by itself and 1. Hence it is a prime number.

That means except 2, all even numbers have more than two divisors(Itself and 1).

You should clarify why this step (the most important step) must be true.

Pi Han Goh - 3 years, 6 months ago

Akash Patalwanshi
Apr 29, 2016

Is the title of question is perfect? May be it is for confusing others!

Yes, it's meant to be that way.

Pi Han Goh - 5 years, 1 month ago

For a second, I thought you were including negative prime numbers.

Sharky Kesa - 5 years, 1 month ago

Prime numbers are always positive by their definition.

akash patalwanshi - 5 years, 1 month ago

@Akash Patalwanshi – You're not exactly right .

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – But by usual definition they are not.

akash patalwanshi - 5 years, 1 month ago

@Akash Patalwanshi – You're right on that one, that is, "by default", prime numbers are all positive numbers. But when we go to some "fancier" math, then the definitions start to break down.

So, if I were to include "negative primes", I would have hinted something along the lines of Gaussian integers .

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – Yes. You are right. But still I prefer the usual definition of prime numbers because if we include negative numbers as well as positive number in definition of prime. Then we get three divisors of a prime number. For example if $-5$ is prime then, there are three divisors of $-5$ that are $1, 5$ and $-5$ itself. In that way then, every number which has three divisors, becomes a prime number and set of prime P looks something different. And in fact then we lost beauty of prime numbers that we have now!

akash patalwanshi - 5 years, 1 month ago

@Akash Patalwanshi – I know where you're coming from. And yes, I totally understand what you're saying. I would just like to point out that mathematicians do sometimes disagree on very basic definitions as well.

Here's another relevant example: What is 0 to the power of 0? (Make sure to click the "reveal the correct answer" section).

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – Thanks for the link.

akash patalwanshi - 5 years, 1 month ago

I've Found More Than One!

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