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Question Number 9049 by Rasheed Soomro last updated on 16/Nov/16

$$\mathrm{Prove}\mathrm{that}\mathrm{every}\mathrm{even}\mathrm{number}\mathrm{can}\mathrm{be}$$$$\mathrm{expressed}\mathrm{as}\mathrm{sum}\mathrm{of}\mathrm{two}\mathrm{primes}\mathrm{or}$$$$\mathrm{give}\mathrm{an}\mathrm{counter}\mathrm{example}.$$

Commented by FilupSmith last updated on 16/Nov/16

$$2n=p_1+p_2$$$$p_n\in \mathbb{P}$$$$$$$$\mathrm{all}\mathrm{primes}\mathrm{are}\mathrm{odd}\mathrm{except}p=2$$$$\therefore 2n=(2a+1)+(2b+1)$$$$2n=2a+2b+2$$$$2n=2(a+b+1)\in \mathbb{E}$$

Commented by FilupSmith last updated on 16/Nov/16

$$\mathrm{if}p=2,2n=p+p=2p=4$$$$2n=4$$

Commented by Rasheed Soomro last updated on 16/Nov/16

$$\mathrm{All}\mathrm{primes}\mathrm{are}\mathrm{odd},\mathrm{but}\mathrm{all}\mathrm{odd}\mathrm{are}$$$$\mathrm{not}\mathrm{necessarily}\mathrm{primes}.$$$$\mathrm{So}2\mathrm{a}+1\mathrm{and}2\mathrm{b}+1\mathrm{here}\mathrm{are}\mathrm{not}$$$$\mathrm{necessarily}\mathrm{prime}.$$

Commented by FilupSmith last updated on 17/Nov/16

$$\mathrm{I}\mathrm{dont}\mathrm{understand}.\mathrm{am}\mathrm{i}\mathrm{wrong}?$$

Commented by mrW last updated on 23/Nov/16

$$\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{famous}{\mathrm{Goldbach}}^{\prime }\mathrm{s}$$$$\mathrm{conjecture},\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{most}$$$$\mathrm{difficult}\mathrm{problems}\mathrm{in}\mathrm{mathematics}.$$

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