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Is it possible to **prove that every** even natural **number greater than** 2 ...

**If Goldbach's conjecture** is **true**, **then** we **could** express **every** sufficiently large ... **If**
N is **odd**, **then** N - **3** is even, and so it **can** be **expressed** as **the sum** of two ... a
**sum** of **three primes**, we're under **no** obligation to use **3**: thus 31 **could** be **7** + 11 +
13. ... How **can** you **prove that every** natural **number greater than** 1 is either **prime**
...

For more information, see Is it possible to **prove that every** even natural **number greater than** 2 ...

Goldbach's conjecture states that every even number > 2 can be expressed as the sum of two prime numbers. The only even prime is 2, so 2 doesn't figure in the primes that Goldbach refers to (how?)

Take n\ >\ 7

Pick a random odd prime number m\ <\ n-2\ ,\ m\ \in P

Now n\ -\ m\ is even and > 2, and hence n\ -m\ =\ p\ +\ q\ where\ p,q\ \in P, by Goldbach's conjecture

Now, n\ =\ m\ +p\ +\ q\ where\ m,p,q\in P

Q.E.D

Let n be an even number greater than 6. Then by Goldbach, there exists primes p and q such that

n = p + q

Where both p and q are odd and at least one is greater than 3. WLOG say q > 3.

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

Applying Goldbach to q + 1 > 4, there exists odd primes r and s such that:

q + 1 = r + s thus,

n + 1 = p + r + s

Where p, r, and s are odd primes.

Finally, note that since n > 6 was an arbitrary even number, n + 1 > 7 is an arbitrary odd number.

I found an answer from www.britannica.com

Christian **Goldbach** | Russian mathematician | Britannica.com

He claimed **that** “**every** number **greater than** 2 is an aggregate of **three prime**
**numbers**. ... (**prime numbers** are now defined as those positive **integers greater**
**than** 1 **that** ... **The** first breakthrough in **the** effort to **prove Goldbach's conjecture**
occurred in ... **odd** natural number **can** be **expressed** as **the sum** of not **more than**
**three** ...

For more information, see Christian **Goldbach** | Russian mathematician | Britannica.com

I found an answer from www.britannica.com

**Number** theory - Pierre de Fermat | Britannica.com

This theorem is one of **the** great tools of modern **number** theory. Fermat
investigated **the** two types of **odd primes**: those **that** are one **more than** a multiple
of 4 and ... Fermat asserted **that any prime** of **the** form 4k + 1 **can** be **written** as **the**
**sum** of .... known as **the Goldbach conjecture**—but acknowledged his inability to
**prove** it.

For more information, see **Number** theory - Pierre de Fermat | Britannica.com

I found an answer from mathworld.wolfram.com

**Goldbach Conjecture** -- from Wolfram MathWorld

are **the sum** of **three odd primes** is called **the** "weak" **Goldbach conjecture**.
Vinogradov (1937ab, 1954) **proved that every** sufficiently large **odd number** is **the**
**sum** of **three** ... **More than** two and a half centuries after **the** original conjecture
was stated, **the** ... **can** be **expressed** as **the sum** of a **prime** plus twice a **prime** is
known as ...

For more information, see **Goldbach Conjecture** -- from Wolfram MathWorld

I found an answer from en.wikipedia.org

**Goldbach's** weak **conjecture** - Wikipedia

In **number** theory, Goldbach's weak conjecture, also known as **the odd Goldbach**
**conjecture**, **the** ternary Goldbach problem, or **the 3**-**primes** problem, states **that**.
**Every odd number greater than** 5 **can** be **expressed** as **the sum** of **three primes**. ...
**Every odd number greater than 7 can** be **expressed** as **the sum** of **three odd** ...

For more information, see **Goldbach's** weak **conjecture** - Wikipedia

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Is there way to know **if ODD number can** be **expressed** as **sum** of two ...

Note **that every prime** is **odd**, with **the** exception of 2, and also note **that** ... **The** first
**odd number greater than** one **that can**'t be **written** as a **sum** of ...

For more information, see Is there way to know **if ODD number can** be **expressed** as **sum** of two ...

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Questions about **the proof that every odd integer** is **the sum** of 5 **primes**

To understand why Theorem 8.2 is sufficient, Tao **write**: On **the** other hand, in [24]
it is shown **that every odd number** larger **than** exp ( 3100 ) is **the sum** of **three** ...

For more information, see Questions about **the proof that every odd integer** is **the sum** of 5 **primes**

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Which **prime numbers can**'t be **represented** as a **sum** of distinct ...

I.e. try to **prove that every integer greater than** 11 **can** be **written** as a **sum** of ... be
**expressed** as (2 + **3** + 5 + **7**), 13 as (2 + 11), but for 2, **3**, 11 no other prime .... **real**-
life conversational skills - so you **can** start speaking a new language in **3** weeks.
..... Be **the** above weak **Goldbach conjecture**, **every odd prime number** is a **sum** ...

For more information, see Which **prime numbers can**'t be **represented** as a **sum** of distinct ...

I found an answer from en.wikipedia.org

**Goldbach's conjecture** - Wikipedia

**Goldbach's conjecture** is one of **the** oldest and best-known unsolved problems in
**number** theory and **all** of mathematics. It states: **Every** even **integer greater than** 2
**can** be **expressed** as **the sum** of two ... **Every integer greater than** 2 **can** be **written**
as **the sum** of **three primes**. ..... "**The** ternary **Goldbach conjecture** is **true**".

For more information, see **Goldbach's conjecture** - Wikipedia