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Prove-that-every-even-number-can-be-expressed-as-sum-of-two-primes-or-give-an-counter-example-




Question Number 9049 by Rasheed Soomro last updated on 16/Nov/16
Prove that every even number can be   expressed as sum of two primes or  give an counter example.
$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{every}\:\mathrm{even}\:\mathrm{number}\:\mathrm{can}\:\mathrm{be}\: \\ $$$$\mathrm{expressed}\:\mathrm{as}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{primes}\:\mathrm{or} \\ $$$$\mathrm{give}\:\mathrm{an}\:\mathrm{counter}\:\mathrm{example}. \\ $$
Commented by FilupSmith last updated on 16/Nov/16
2n=p_1 +p_2   p_n ∈P     all primes are odd except p=2  ∴2n=(2a+1)+(2b+1)  2n=2a+2b+2  2n=2(a+b+1)∈E
$$\mathrm{2}{n}={p}_{\mathrm{1}} +{p}_{\mathrm{2}} \\ $$$${p}_{{n}} \in\mathbb{P} \\ $$$$\: \\ $$$$\mathrm{all}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{odd}\:\mathrm{except}\:{p}=\mathrm{2} \\ $$$$\therefore\mathrm{2}{n}=\left(\mathrm{2}{a}+\mathrm{1}\right)+\left(\mathrm{2}{b}+\mathrm{1}\right) \\ $$$$\mathrm{2}{n}=\mathrm{2}{a}+\mathrm{2}{b}+\mathrm{2} \\ $$$$\mathrm{2}{n}=\mathrm{2}\left({a}+{b}+\mathrm{1}\right)\in\mathbb{E} \\ $$
Commented by FilupSmith last updated on 16/Nov/16
if p=2, 2n=p+p=2p=4  2n=4
$$\mathrm{if}\:{p}=\mathrm{2},\:\mathrm{2}{n}={p}+{p}=\mathrm{2}{p}=\mathrm{4} \\ $$$$\mathrm{2}{n}=\mathrm{4} \\ $$
Commented by Rasheed Soomro last updated on 16/Nov/16
All primes are odd,but all odd are  not necessarily primes.  So 2a+1 and 2b+1 here are not  necessarily prime.
$$\mathrm{All}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{odd},\mathrm{but}\:\mathrm{all}\:\mathrm{odd}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{necessarily}\:\mathrm{primes}. \\ $$$$\mathrm{So}\:\mathrm{2a}+\mathrm{1}\:\mathrm{and}\:\mathrm{2b}+\mathrm{1}\:\mathrm{here}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{necessarily}\:\mathrm{prime}. \\ $$
Commented by FilupSmith last updated on 17/Nov/16
I dont understand. am i wrong?
$$\mathrm{I}\:\mathrm{dont}\:\mathrm{understand}.\:\mathrm{am}\:\mathrm{i}\:\mathrm{wrong}? \\ $$
Commented by mrW last updated on 23/Nov/16
This is the famous Goldbach′s  conjecture, one of the most  difficult problems in mathematics.
$$\mathrm{This}\:\mathrm{is}\:\mathrm{the}\:\mathrm{famous}\:\mathrm{Goldbach}'\mathrm{s} \\ $$$$\mathrm{conjecture},\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{most} \\ $$$$\mathrm{difficult}\:\mathrm{problems}\:\mathrm{in}\:\mathrm{mathematics}. \\ $$

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