# Expressing Natural Numbers as Sum of Primes

• metrictensor
In summary, according to Bertrand's postulate, it is possible to express all natural numbers greater than 2 as the sum of N unique prime numbers.

#### metrictensor

Is it possible to express all natural numbers greater than 2 as the sum of N unique prime numbers? For example, 6 = 2 + 3 and 18 = 13 + 5.

6=2+3 does it?

What about the smallest natural greater than 2 that isn't a prime? isn't that a counter example too?

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matt grime said:
6=2+3 does it?

What about the smallest natural greater than 2 that isn't a prime? isn't that a counter example too?
I forgot to mention that 1 and 2 can be used in the sum. I made a mistake. Not 6 = 2 + 3, 5 = 2 + 3. Once you define 1 and 2 you get 3. From 3+1 you get 4 and so on. 5=2+3...

So you are NOT talking about a "sum of primes" because 1 is not prime.

Now I'm a little perplexed. 2 is a prime. Anyway, the result follows quite easily from Russell's postulate:

For every n>1 there is a prime number p such that n<p<2n.

So, given m an integer, if it is even pick an prime in the range m/2 to m, and if it is odd pick a prime in the range (m+1)/2 to m+1, call this prime p(1).

Then m-p(1) < m/2, and we proceed by indcuction - the next prime we pick must be distinct from p(1) since it must be less than m/2, and p(1) is greater than m/2.

We just need to show how to write the sum for "small m" to get the full proof, ie m=1,2,3 which yo'uve done.

Matt grime: the result follows quite easily from Russell's postulate.

Yeah? As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes.http://mathworld.wolfram.com/GoldbachConjecture.html

So you are saying the Goldbach Conjecture has been proven?

Schnirelman (1939) proved that every even number can be written as the sum of not more than 300 000 primes (Dunham 1990)

What if we added up the first 300 002 primes?

Icebreaker said:
What if we added up the first 300 002 primes?
What is that supposed to mean :uhh: ?

Nevermind.

robert Ihnot said:
Matt grime: the result follows quite easily from Russell's postulate.

Yeah? As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes.http://mathworld.wolfram.com/GoldbachConjecture.html

So you are saying the Goldbach Conjecture has been proven?

Of course not, but the OP didn't ask for the sum of TWO primes, he asked for the sum to be expressed as *some* sum of N distinct primes, allowing for 1 to be included.

The possibly dodgy assumption I made was that it wasn't supposed to be a fixed N, which seems reasonable, given the other hidden assumptions in the post, after all it would be required that N=1 or 2 otherwise, which we know to be false and bloody hard respectively.

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Well, whatever it should be called, it states given any natural n greater than 2 there is a prime p satisfying n<p<2n, or similar.

matt grime said:
Well, whatever it should be called, it states given any natural n greater than 2 there is a prime p satisfying n<p<2n, or similar.
It has been proved there exists a prime for any natural number n > 2 there exists a prime (and I'm really dragging this one from memory) p such that:

$$n - n^{\frac{23}{42}} < p < n$$

Which is quite a bit stronger . Although I'm not sure how useful.

Here's the formula on the Wolfram site

Zurtex said:
It has been proved there exists a prime for any natural number n > 2 there exists a prime (and I'm really dragging this one from memory) p such that:

$$n - n^{\frac{23}{42}} < p < n$$

Which is quite a bit stronger . Although I'm not sure how useful.

http://functions.wolfram.com/NumberTheoryFunctions/Prime/29/0003/

## 1. What does it mean to express a natural number as a sum of primes?

Expressing a natural number as a sum of primes means finding a combination of prime numbers that add up to the given number. This is also known as the Prime Factorization of a number.

## 2. Why is it important to express natural numbers as sums of primes?

Expressing natural numbers as sums of primes is important because it helps us understand the fundamental building blocks of numbers. It also has applications in cryptography and number theory.

## 3. How do you express a natural number as a sum of primes?

To express a natural number as a sum of primes, you can use the process of prime factorization. This involves breaking down the number into its prime factors and then rearranging them to form the sum.

## 4. Can every natural number be expressed as a sum of primes?

Yes, every natural number can be expressed as a sum of primes. This is known as the Fundamental Theorem of Arithmetic, which states that every natural number has a unique prime factorization.

## 5. Are there any patterns or rules for expressing natural numbers as sums of primes?

There are no specific patterns or rules for expressing natural numbers as sums of primes. However, there are some strategies and techniques, such as using trial division or the Sieve of Eratosthenes, that can make the process easier and more efficient.