Maths help needed

FirePig

8 Apr 08, 23:25

Help me prove that any even integer greater than 2 is the sum of two prime numbers. Thanks.

Xarxos

9 Apr 08, 04:07

No current solution. Don't be a dick.

http://en.wikipedia.org/wiki/Goldbach_conjecture

FirePig

10 Apr 08, 13:32

Okay make it easier.

Prove that the sum of two prime numbers, p + q is an even integer where p and q are primes not equal to 2,

LatecomerX

10 Apr 08, 13:38

Originally posted by FirePig:

Okay make it easier.

Prove that the sum of two prime numbers, p + q is an even integer where p and q are primes not equal to 2,

Since all prime numbers except 2 is odd, and any sum of two odd integers will always result in an even integer, hence p + q will always be an even integer.

FirePig

10 Apr 08, 13:42

Originally posted by LatecomerX:

Since all prime numbers except 2 is odd, and any sum of two odd integers will always result in an even integer, hence p + q will always be an even integer.

Yeah i know it is easy. But can you pls go deeper and prove that all prime numbers except 2 is odd. And prove also that any sum of two odd integers is even.

^tamago^

10 Apr 08, 13:51

Originally posted by FirePig:

And prove also that any sum of two odd integers is even.

For any two odd numbers p₁ and p₂ is real,
p₁ ≡ 1 mod 2
p₂ ≡ 1 mod 2

p₁ + p₂ ≡ 1+1 mod 2 ≡ 2 mod 2 ≡ 0 mod 2

∴ The sum of 2 odd numbers is even.

weewee

10 Apr 08, 16:14

Originally posted by FirePig:

Yeah i know it is easy. But can you pls go deeper and prove that all prime numbers except 2 is odd. And prove also that any sum of two odd integers is even.

All prime numbers except 2 is odd. Suppose otherwise. That is there is a prime number p other than 2 which is even. Then p can be divided by 2 which means that p is not prime. This contradicts the assumption. Hence such a p does not exist.

Let the two odd integers be p and q. Odd intergers can be expressed in the form 2n+1 where n is a particular integer. eg p = 2n + 1, q = 2k +1 where n and k is an integer.

p + q = 2n + 1 + 2k + 1 = 2n + 2k + 2 = 2 ( n + k + 1)

Since n + k + 1 is an integer, p +q is an even integer.

eagle

10 Apr 08, 20:26

Originally posted by FirePig:

Yeah i know it is easy. But can you pls go deeper and prove that all prime numbers except 2 is odd. And prove also that any sum of two odd integers is even.

actually just think... is there any other positive even integer greater than 2 that is not divisible by 2?

FirePig

10 Apr 08, 23:10

Hi all. I believe this conjecture cannot be proven as prime number is an unnatural definition invented by men.

4 = 2 + 2

6 = 3 + 3

8 = 3 + 5

10 = 3 + 7 = 5 + 5

12 = 5 + 7

14 = 3 + 11 = 7 + 7

For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, N. Pipping in 1938 laboriously verified the conjecture up to $n \leq 10^5$ . With the advent of computers, many more small values of n have been checked; T. Oliveira e Silva is running a distributed computer search that has verified the conjecture for $n \leq 10^{18}$ .

Statistical considerations which focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture (in both the weak and strong forms) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes.

Number of ways to write an even number n as the sum of two primes (4 ≤ n ≤ 1,000,000)