Question

Goldbach primer

Answer 1

ok so can u write the definition that you know about Goldbach ?

Answer 2

that would help me to explain when i know what you bassiclly know xD

Answer 3

Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics.

http://en.wikipedia.org/wiki/Goldbach%27s_conjecture

Answer 4

is that good ?

Answer 5

ok first line is enough
\(\text{every even integer greater than 2 can be expressed as the sum of two primes. }\)

Answer 6

that makes us think of 2 things :-
1- sum of 2 primes can represent unique even integer
2- an even number can be represented using at least one group of 2 primes

agree so far ?

Answer 7

yes

Answer 8

yes but they may not cover all even numbers

Answer 9

thats a good notation but note
first i said they are infinite , however what im trying to do
is seeing how much they can recover

Answer 10

ok

Answer 11

remember my question
\(\Large \Large \text {let P be the set of first nth odd primes } \\\Large P= \left \{ p_1,p2,....p_n \right\} \\ \Large \text{ and } \Large S= \left \{ p_i+p_j :\forall i,j\leq n \right \} \\ \Large \text{ then , show that there exist x(n) } \\ \Large \left \{ 6,8,10,.....,2p_{x(n)} \right \}\subseteq S \text { note that }\\ \Large
x(n) \text{is a function of n } \\ \Large \text{with Range
and domain of N and } x(n)<n
\\ \text{ PS :- } \left \{ 6,8,10,.....,2p_{x(n)} \right \} =\left \{ 2\times 3 ,2\times 4,..., 2(p_{x(n ) }-1) ,2p_{x(n)} \right \}\)

Answer 12

S is the group of all even that it can cover , ( agree ?)

Answer 13

dont read it all i'll just explain bit by bit

Answer 14

i see lot of typoes in that

Answer 15

ugh i see why u confused
so a finite group of primes can represent a finite group of even numbers

sorry for that :(

Answer 16

i was confused about goldbach from the start
typoes will only make it worse

Answer 17

hmm yeah sorry about it i meant to say
-"so a finite group of primes can represent a finite group of even numbers "

so for ur question

-"yes but they may not cover all even numbers"

-first i said they are "finite" , however what im trying to do
is seeing how much they can recover

Answer 18

\[\Large \Large \text {let P be the set of first n odd primes } \\\Large P= \left \{ p_1,p_2,\ldots,p_n \right\} \\ \Large \text{ and } \Large S= \left \{ p_i+p_j : i,j\leq n \right \} \\ \Large \text{ then , show that there exists x(n) } \\ \Large \left \{ 6,8,10,.....,2p_{x(n)} \right \}\subseteq S \]

Answer 19

i get the definition of P and S sets
i dont get x(n) and remaining part

Answer 20

ok cool

Answer 21

x(n) is just an increasing function , with respect to n
properties
1- not constant
2- when n goes to infinity x(n) goes to infinity
3- domain and range of x(n) is natural numbers

Answer 22

ok

Answer 23

so S as i said is the set of all evens can be represented by P group
but when we order it , it might be discontinues at some values i'll show u an example

Answer 24

\(\Large P=\left \{ 3,5,7,11 \right \}\\ \Large S=\left \{ 6,8,10,12,14,16,18,{\color{Red} 22} \right \} \)

Answer 25

so 20 is missing here ?

Answer 26

so if we define a set
V={6,8,.......,2k} k>2

when k=9 then
\(V \subseteq S\) and V would be the largest set contained in S can be of this form

Answer 27

yes , but i focused on discontinuity at 2 its discontinues (right ?)

Answer 28

at 22 *

Answer 29

agree so far ?

Answer 30

if not tell me ur doubts

Answer 31

now ,in general we cant know k that represent represent V which is the largest even ordered set contained in S since we get S with P elements only

Answer 32

V is a set of first k positive even integers ?

Answer 33

yes contained in S

Answer 34

ok no i see why it is interesting

Answer 35

since V is mystery for us (in General )
why cant we find another set contained in V ( so ovc it would be contained in S )
but have the largest element of 2*p such that p is prime ?

Answer 36

*now

Answer 37

like for my example above
p={3,5,7,11}
S={6,8,10,12,14,16,18,22}
V= {6,8,10,12,14,16,18}
and let G be the set im talking about
G= {6,8,10,12,14 }, 14=2*7

Answer 38

{6,8,10,12,14}

Answer 39

yes !

Answer 40

so in P set we have 4 elements ( n=4)
and number 7 have the 3ed position
so
x(4)=3=4-1 (in this case )

Answer 41

ok what other doubts u have so far ?

Answer 42

i'll show u how this is equivelent to Goldbatch if u want to

Answer 43

```
so in P set we have 4 elements ( n=4)
and number 7 have the 3ed position
so
x(4)=3=4-1 (in this case )
```

Answer 44

i dont get this

Answer 45

P={3,5,7,11}

P has 4 elements so that gives n = 4

Answer 46

yes

Answer 47

and G set have 2*7
if we mapped 7 to P 7 have the 3ed position ok ?

Answer 48

instead of knowing k that evaluate V
we need to know what is the prime that gives us G

Answer 49

??

Answer 50

\(x(n) = n-1\) works always ?

Answer 51

naa i talked about the existence of having such a function , so according to goldbach there exist , this is what im also trying to show
x(n)=n-1 fail with n =5 i guess

Answer 52

how ever , gap btw primes become larger
there is small gap in small number but if we considered huge numbers gap sometimes is huge as well from here i had problem of choosing function
since i need x(n) slowly increasing :)
so with small n linear work with larger sqrt work with larger log work

Answer 53

i hope u gt it any doubt im here , u can ask

Answer 54

nice