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metrictensor
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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.
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...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?
Schnirelman (1939) proved that every even number can be written as the sum of not more than 300 000 primes (Dunham 1990)
What is that supposed to meanIcebreaker said:What if we added up the first 300 002 primes?
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?
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: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.
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:
[tex]n - n^{\frac{23}{42}} < p < n[/tex]
Which is quite a bit stronger. Although I'm not sure how useful.
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.
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.
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.
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.
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.