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Question Number 3595 by Filup last updated on 16/Dec/15
I just thought of something I am curious in figuring out. All integer numbers can be made up by prime factors. That is: n=p_1 ×p_2 ×...×p_i n∈Z p_k ∈P Are there an inifinite number of numbers that are the sum of prime numbers? That is: P=p_1 +p_2 +...+p_i P,p_k ∈P For example: 2+3=5 2+3+3+5=13 etc. Are all of these special primes odd? What else can we work out?
$$\mathrm{I}\:\mathrm{just}\:\mathrm{thought}\:\mathrm{of}\:\mathrm{something}\:\mathrm{I}\:\mathrm{am}\:\mathrm{curious} \\ $$$$\mathrm{in}\:\mathrm{figuring}\:\mathrm{out}. \\ $$$$ \\ $$$$\mathrm{All}\:\mathrm{integer}\:\mathrm{numbers}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\:\mathrm{up}\:\mathrm{by} \\ $$$${prime}\:{factors}.\:\mathrm{That}\:\mathrm{is}: \\ $$$${n}={p}_{\mathrm{1}} ×{p}_{\mathrm{2}} ×…×{p}_{{i}} \\ $$$${n}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:{p}_{{k}} \in\mathbb{P} \\ $$$$ \\ $$$$\mathrm{Are}\:\mathrm{there}\:\mathrm{an}\:\mathrm{inifinite}\:\mathrm{number}\:\mathrm{of}\:\mathrm{numbers} \\ $$$$\mathrm{that}\:\mathrm{are}\:\mathrm{the}\:{sum}\:\mathrm{of}\:{prime}\:{numbers}? \\ $$$$\mathrm{That}\:\mathrm{is}: \\ $$$${P}={p}_{\mathrm{1}} +{p}_{\mathrm{2}} +…+{p}_{{i}} \\ $$$${P},{p}_{{k}} \in\mathbb{P} \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{example}: \\ $$$$\mathrm{2}+\mathrm{3}=\mathrm{5} \\ $$$$\mathrm{2}+\mathrm{3}+\mathrm{3}+\mathrm{5}=\mathrm{13} \\ $$$${etc}. \\ $$$$ \\ $$$$\mathrm{Are}\:\mathrm{all}\:\mathrm{of}\:\mathrm{these}\:\mathrm{special}\:\mathrm{primes}\:\mathrm{odd}? \\ $$$$\mathrm{What}\:\mathrm{else}\:\mathrm{can}\:\mathrm{we}\:\mathrm{work}\:\mathrm{out}? \\ $$
Commented by Yozzii last updated on 16/Dec/15
I think your question is a case of the abc conjecture which hasn′t been neatly proven yet.
$${I}\:{think}\:{your}\:{question}\:{is}\:{a}\:{case} \\ $$$${of}\:{the}\:{abc}\:{conjecture}\:{which}\:{hasn}'{t} \\ $$$${been}\:{neatly}\:{proven}\:{yet}.\: \\ $$
Commented by Yozzii last updated on 16/Dec/15
abc conjecture: For any infinitesimal ε>0 there exists a constant C_ε such that, for a,b,c being coprime satisfying a+b=c, the inequality max(∣a∣,∣b∣,∣c∣)≤C_ε Π_(p∣abc) p^(1+ε) holds, where p∣abc is the product over primes p which divide the product abc.
$${abc}\:{conjecture}: \\ $$$$ \\ $$$${For}\:{any}\:{infinitesimal}\:\epsilon>\mathrm{0}\:{there} \\ $$$${exists}\:{a}\:{constant}\:{C}_{\epsilon} \:{such}\:{that},\:{for} \\ $$$${a},{b},{c}\:{being}\:{coprime}\:{satisfying} \\ $$$$\:\:\:\:\:{a}+{b}={c}, \\ $$$${the}\:{inequality} \\ $$$$\:\:\:\:\:{max}\left(\mid{a}\mid,\mid{b}\mid,\mid{c}\mid\right)\leqslant{C}_{\epsilon} \underset{{p}\mid{abc}} {\prod}{p}^{\mathrm{1}+\epsilon} \\ $$$${holds},\:{where}\:{p}\mid{abc}\:{is}\:{the}\:{product}\:{over} \\ $$$${primes}\:{p}\:{which}\:{divide}\:{the}\:{product}\:{abc}. \\ $$
Commented by Filup last updated on 16/Dec/15
Oh? What is that?
$$\mathrm{Oh}?\:\mathrm{What}\:\mathrm{is}\:\mathrm{that}? \\ $$
Commented by Filup last updated on 16/Dec/15
if P∈E: Σ_(k=1) ^i p_k =P P∈E ∴Σ=2i ∵if P∈E, P=2+2+2+... (even = even + even) (2 is only even prime) P=2i i∈Z ∴P∉P ∴all special primes are odd correct?
$$\mathrm{if}\:{P}\in\mathbb{E}: \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{i}} {\sum}}{p}_{{k}} ={P}\:\:\:\:\:\:\:\:\:{P}\in\mathbb{E} \\ $$$$\therefore\Sigma=\mathrm{2}{i}\:\:\:\because\mathrm{if}\:{P}\in\mathbb{E},\:{P}=\mathrm{2}+\mathrm{2}+\mathrm{2}+… \\ $$$$\left({even}\:=\:{even}\:+\:{even}\right) \\ $$$$\left(\mathrm{2}\:\mathrm{is}\:\mathrm{only}\:\mathrm{even}\:\mathrm{prime}\right) \\ $$$${P}=\mathrm{2}{i}\:\:\:\:{i}\in\mathbb{Z} \\ $$$$\therefore{P}\notin\mathbb{P} \\ $$$$ \\ $$$$\therefore\mathrm{all}\:\mathrm{special}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{odd} \\ $$$$\mathrm{correct}? \\ $$
Commented by prakash jain last updated on 16/Dec/15
I thought what Filup is asking is: Can every prime (≥5) be expressed as sum of more than 2 primes: 5=2+3 7=2+2+3 11=7+2+2
$$\mathrm{I}\:\mathrm{thought}\:\mathrm{what}\:\mathrm{Filup}\:\mathrm{is}\:\mathrm{asking}\:\mathrm{is}: \\ $$$$\mathrm{Can}\:\mathrm{every}\:\mathrm{prime}\:\left(\geqslant\mathrm{5}\right)\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{as}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{more}\:\mathrm{than}\:\mathrm{2}\:\mathrm{primes}: \\ $$$$\mathrm{5}=\mathrm{2}+\mathrm{3} \\ $$$$\mathrm{7}=\mathrm{2}+\mathrm{2}+\mathrm{3} \\ $$$$\mathrm{11}=\mathrm{7}+\mathrm{2}+\mathrm{2} \\ $$
Commented by Filup last updated on 16/Dec/15
Yes, that is a part of what im asking
$$\mathrm{Yes},\:{that}\:\mathrm{is}\:\mathrm{a}\:\mathrm{part}\:\mathrm{of}\:\mathrm{what}\:\mathrm{im}\:\mathrm{asking} \\ $$
Commented by Filup last updated on 16/Dec/15
I have proven that all special primes are odd. But can all primes be written by the sum of other primes??
$$\mathrm{I}\:\mathrm{have}\:\mathrm{proven}\:\mathrm{that}\:\mathrm{all}\:\mathrm{special}\:\mathrm{primes}\:\mathrm{are} \\ $$$$\mathrm{odd}. \\ $$$$\mathrm{But}\:\mathrm{can}\:{all}\:\mathrm{primes}\:\mathrm{be}\:\mathrm{written}\:\mathrm{by}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{other}\:\mathrm{primes}?? \\ $$
Commented by 123456 last updated on 16/Dec/15
if there is a infinite number of twin primes then there a infinite number of primes that can be write by sum of two or more prime number
$$\mathrm{if}\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{infinite}\:\mathrm{number}\:\mathrm{of}\:\mathrm{twin}\:\mathrm{primes} \\ $$$$\mathrm{then}\:\mathrm{there}\:\mathrm{a}\:\mathrm{infinite}\:\mathrm{number}\:\mathrm{of}\:\mathrm{primes} \\ $$$$\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{write}\:\mathrm{by}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{or}\:\mathrm{more} \\ $$$$\mathrm{prime}\:\mathrm{number} \\ $$
Commented by 123456 last updated on 16/Dec/15
add a 3 and you can generate all numbers ≥2 2i+3j 00 02 04 06 08 10 … 03 05 07 09 11 13 … 06 08 10 12 14 16 … 09 11 13 15 17 19 … 12 14 16 18 20 22 … 15 17 19 21 23 25 … ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
$$\mathrm{add}\:\mathrm{a}\:\mathrm{3}\:\mathrm{and}\:\mathrm{you}\:\mathrm{can}\:\mathrm{generate}\:\mathrm{all}\:\mathrm{numbers}\:\geqslant\mathrm{2} \\ $$$$\mathrm{2}{i}+\mathrm{3}{j} \\ $$$$\mathrm{00}\:\mathrm{02}\:\mathrm{04}\:\mathrm{06}\:\mathrm{08}\:\mathrm{10}\:\ldots \\ $$$$\mathrm{03}\:\mathrm{05}\:\mathrm{07}\:\mathrm{09}\:\mathrm{11}\:\mathrm{13}\:\ldots \\ $$$$\mathrm{06}\:\mathrm{08}\:\mathrm{10}\:\mathrm{12}\:\mathrm{14}\:\mathrm{16}\:\ldots \\ $$$$\mathrm{09}\:\mathrm{11}\:\mathrm{13}\:\mathrm{15}\:\mathrm{17}\:\mathrm{19}\:\ldots \\ $$$$\mathrm{12}\:\mathrm{14}\:\mathrm{16}\:\mathrm{18}\:\mathrm{20}\:\mathrm{22}\:\ldots \\ $$$$\mathrm{15}\:\mathrm{17}\:\mathrm{19}\:\mathrm{21}\:\mathrm{23}\:\mathrm{25}\:\ldots \\ $$$$\vdots\:\vdots\:\vdots\:\vdots\:\vdots\:\vdots\:\ddots \\ $$
Commented by prakash jain last updated on 16/Dec/15
So all numbers (including primes) can be written a sum of 2 or more primes.
$$\mathrm{So}\:\mathrm{all}\:\mathrm{numbers}\:\left(\mathrm{including}\:\mathrm{primes}\right)\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{written}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{2}\:\mathrm{or}\:\mathrm{more}\:\mathrm{primes}. \\ $$

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