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Question Number 3595 by Filup last updated on 16/Dec/15

$$\mathrm{I}\mathrm{just}\mathrm{thought}\mathrm{of}\mathrm{something}\mathrm{I}\mathrm{am}\mathrm{curious}$$$$\mathrm{in}\mathrm{figuring}\mathrm{out}.$$$$$$$$\mathrm{All}\mathrm{integer}\mathrm{numbers}\mathrm{can}\mathrm{be}\mathrm{made}\mathrm{up}\mathrm{by}$$$$primefactors.\mathrm{That}\mathrm{is}:$$$$n=p_1\times p_2\times \dots \times p_i$$$$n\in \mathbb{Z}{p}_k\in \mathbb{P}$$$$$$$$\mathrm{Are}\mathrm{there}\mathrm{an}\mathrm{inifinite}\mathrm{number}\mathrm{of}\mathrm{numbers}$$$$\mathrm{that}\mathrm{are}\mathrm{the}sum\mathrm{of}primenumbers?$$$$\mathrm{That}\mathrm{is}:$$$$P=p_1+p_2+\dots +p_i$$$$P,p_k\in \mathbb{P}$$$$$$$$\mathrm{For}\mathrm{example}:$$$$2+3=5$$$$2+3+3+5=13$$$$etc.$$$$$$$$\mathrm{Are}\mathrm{all}\mathrm{of}\mathrm{these}\mathrm{special}\mathrm{primes}\mathrm{odd}?$$$$\mathrm{What}\mathrm{else}\mathrm{can}\mathrm{we}\mathrm{work}\mathrm{out}?$$

Commented by Yozzii last updated on 16/Dec/15

$$Ithinkyourquestionisacase$$$$oftheabcconjecturewhich{hasn}^{\prime }t$$$$beenneatlyprovenyet.$$

Commented by Yozzii last updated on 16/Dec/15

$$abcconjecture:$$$$$$$$Foranyinfinitesimalϵ>0there$$$$existsaconstant{C}_{ϵ}suchthat,for$$$$a,b,cbeingcoprimesatisfying$$$$a+b=c,$$$$theinequality$$$$max(\mid a\mid ,\mid b\mid ,\mid c\mid )⩽C_{ϵ}\prod _{p\mid abc}{p}^{1+ϵ}$$$$holds,wherep\mid abcistheproductover$$$$primespwhichdividetheproductabc.$$

Commented by Filup last updated on 16/Dec/15

$$\mathrm{Oh}?\mathrm{What}\mathrm{is}\mathrm{that}?$$

Commented by Filup last updated on 16/Dec/15

$$\mathrm{if}P\in \mathbb{E}:$$$$\underset{k=1}{\sum ^i}{p}_k=PP\in \mathbb{E}$$$$\therefore \mathrm{\Sigma }=2i\because \mathrm{if}P\in \mathbb{E},P=2+2+2+\dots$$$$(even=even+even)$$$$\left(2\mathrm{is}\mathrm{only}\mathrm{even}\mathrm{prime}\right)$$$$P=2ii\in \mathbb{Z}$$$$\therefore P\notin \mathbb{P}$$$$$$$$\therefore \mathrm{all}\mathrm{special}\mathrm{primes}\mathrm{are}\mathrm{odd}$$$$\mathrm{correct}?$$

Commented by prakash jain last updated on 16/Dec/15

$$\mathrm{I}\mathrm{thought}\mathrm{what}\mathrm{Filup}\mathrm{is}\mathrm{asking}\mathrm{is}:$$$$\mathrm{Can}\mathrm{every}\mathrm{prime}(⩾5)\mathrm{be}\mathrm{expressed}\mathrm{as}\mathrm{sum}\mathrm{of}$$$$\mathrm{more}\mathrm{than}2\mathrm{primes}:$$$$5=2+3$$$$7=2+2+3$$$$11=7+2+2$$

Commented by Filup last updated on 16/Dec/15

$$\mathrm{Yes},that\mathrm{is}\mathrm{a}\mathrm{part}\mathrm{of}\mathrm{what}\mathrm{im}\mathrm{asking}$$

Commented by Filup last updated on 16/Dec/15

$$\mathrm{I}\mathrm{have}\mathrm{proven}\mathrm{that}\mathrm{all}\mathrm{special}\mathrm{primes}\mathrm{are}$$$$\mathrm{odd}.$$$$\mathrm{But}\mathrm{can}all\mathrm{primes}\mathrm{be}\mathrm{written}\mathrm{by}\mathrm{the}$$$$\mathrm{sum}\mathrm{of}\mathrm{other}\mathrm{primes}??$$

Commented by 123456 last updated on 16/Dec/15

$$\mathrm{if}\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{infinite}\mathrm{number}\mathrm{of}\mathrm{twin}\mathrm{primes}$$$$\mathrm{then}\mathrm{there}\mathrm{a}\mathrm{infinite}\mathrm{number}\mathrm{of}\mathrm{primes}$$$$\mathrm{that}\mathrm{can}\mathrm{be}\mathrm{write}\mathrm{by}\mathrm{sum}\mathrm{of}\mathrm{two}\mathrm{or}\mathrm{more}$$$$\mathrm{prime}\mathrm{number}$$

Commented by 123456 last updated on 16/Dec/15

$$\mathrm{add}\mathrm{a}3\mathrm{and}\mathrm{you}\mathrm{can}\mathrm{generate}\mathrm{all}\mathrm{numbers}⩾2$$$$2i+3j$$$$000204060810\dots$$$$030507091113\dots$$$$060810121416\dots$$$$091113151719\dots$$$$121416182022\dots$$$$151719212325\dots$$$$⋮⋮⋮⋮⋮⋮\ddots$$

Commented by prakash jain last updated on 16/Dec/15

$$\mathrm{So}\mathrm{all}\mathrm{numbers}\left(\mathrm{including}\mathrm{primes}\right)\mathrm{can}\mathrm{be}$$$$\mathrm{written}\mathrm{a}\mathrm{sum}\mathrm{of}2\mathrm{or}\mathrm{more}\mathrm{primes}.$$

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