Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

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.

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com