60% Of The Time, It Works Every Time

Number Theory Level 1

$\begin{aligned} 4 &=& 2+2 \\ 5&=&2+3 \\ 6&=&3+3 \end{aligned}$

The above shows that the first 3 positive integers larger than 3 can be expressed as the sum of 2 prime numbers .

Is it true that we can express any positive integer larger than 3 as the sum of 2 prime numbers?

Yes, it is true No, it is not true

2 solutions

Chew-Seong Cheong
May 22, 2016

Relevant wiki: Prime Numbers

All primes except 2 are odd. If the claim of all positive integer larger than 3 is a sum of two primes is true, then for any prime larger than 3 to be odd, one of the two primes must be 2. This also means that subsequent primes must be 2 apart. It is true for 5 and 7 but not true for 11, 17, 23 and so on. Therefore, $\boxed{\text{no, it is not true.}}$ .

Great! Now, is it true that an even number can be expressed as two prime numbers?

Pi Han Goh - 5 years ago

It is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It is called Goldbach's conjecture , which states:

Every even integer greater than 2 can be expressed as the sum of two primes.

Chew-Seong Cheong - 5 years ago

False, 2 = 1+1 where 1 is not prime and 2 = 0 + 2 where 0 is not prime. 0 is even too which can only be expressed as 0+0 where 0 is not prime.

Ashish Menon - 5 years ago

Actually, the original Goldbach's conjecture states that every number can be expressed as the sum of 3 prime numbers; it was later that Euler changed the Goldbach's conjecture for even numbers.

Mateo Matijasevick - 5 years ago

but the question asked any number above 3 can be expressed as sum of two primes...what you are telling is basically Goldbach's conjecture...so according to this specific question the answer must be true...

Subhra Shankha Bhattacherjee - 5 years ago

You misread the question. The answer is "No, it is not true". Plus, it has nothing to do with Goldbach's conjecture.

Pi Han Goh - 5 years ago

sorry my bad..

Subhra Shankha Bhattacherjee - 5 years ago

Ashish Menon
May 22, 2016

Relevant wiki: Prime Numbers

Check for first few values.
4, 5 and 6 are ruled out as mentioned in the question. 7 = 5 + 2 so its is ruled out too. 8 = 5 + 3 so it is ruled out too. 9 = 2 + 7 so it is ruled out too. 10 = 7 + 3 so it is ruled out too.
Now, 11 can be written as 0 + 11 but 0 is not prme, 1 + 10 but both are not prime, 2+ 9 but 9 is not prime, 3 + 8 but 8 is not prime, 4 + 7 but 7 is not prime, 5 + 6 but six is not prime. There is no other way of expressing 11 as a sum of two integers. So 11 cannot be expressed as the sum of any two primes. So the answer is $\color{#69047E}{\boxed{\text{No, its not true}}}$ .

Not really a solution? This is indeed a solution! You just showed us a counterexample.

This is correct!

Pi Han Goh - 5 years ago

Um.. I thought its not because i didnt prove it by induction.

Ashish Menon - 5 years ago

Induction? No, it's not applicable here.

If you want to disprove a universal statement (like my claim), then you just need to give a counterexample: something that doesn't work.

For example: Is all prime numbers odd? My answer would be no, because 2 is a prime number that is not an odd number, so I've found something that doesn't fit this category. So my claim of "All prime numbers are odd" is wrong.

Pi Han Goh - 5 years ago

@Pi Han Goh – Ok haha, I removed it. Thanks! :)

Ashish Menon - 5 years ago

But what does the Goldbach conjecture say.

Aaron Jerry Ninan - 4 years, 10 months ago

60% Of The Time, It Works Every Time

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