IIT JEE , AIEEE Forum - Mathematics , Algebra - 5 SALUTES IF U CAN CRACK THIS...............

parismulye

PROVE THAT ANY EVEN NATURAL NO.CAN BE EXPRESSED IN THE FORM OF ADDITTION OF 2 PRIME NO.S(EXCEPT 2)

COME ON.......TRY AS HARD AS U CAN

fused_bulb

Hi....

since u r a gal....i will talk a bit politely...

see....there is no point in being oversmart and giving us all

the " GOLDBACH ' S CONJECTURE " .....to solve..

this question seems quite simple but has been unsolved for centuries...

i am saying all this because i dont want anybody to waste time on this question....

n i bet u....if anybody solves this ........he would be among the top mathematicians in the world ...which i doubt ...anybody is capable of here..( at this age )..

nikunj_vm

except 2 all prime no have difference of 2or 4or 6etc

eg.3,5,7,11,13,17,19,23...

so we can can make any even natural no . by combining 2 prime nos

am i correct

chimanshu_007

see i m jst trying

all prime no.s are odd , bcoz if they were even , they wud be divided by 2 hence they wud not be called as prime

and we all know sum of any 2 odd no.s is an even no.

so ANY EVEN NATURAL NO.CAN BE EXPRESSED IN THE FORM OF ADDITTION OF 2 PRIME NO.S(EXCEPT 2)

saarika

i think nikunj is ryt

even no.s have a diff of 2

nd prime nos have a diff of 2,4,6..........hence if one even no can be made by addition of two prime numbers then all of the even no.s can be made

eg

8= 3+5 sum of two primes

10= 8+2 =3+5+2=(3+2)+5 =5+5 sum of two primes

or =3+(5+2)= 3+7 sum of two primes

12 = 8+4 =(3+4)+5=7+5 sum of two primes.......

and so on

nd himanshu all prime numbers are odd (except 2) but all oddnos are not primes

two odd no's will give even no on addition but the odd nos may not be primes

debabratanag99

I think people here are making a mistake............

It is true that if two prime numbers are added they will always result in an even number BUT that does not prove the reverse that ALL even numbers can be produced the same way...........!!!!!!!!!!!

This is known as......................
Goldbach's conjecture ...........................

THE PROBABLE PROOF OF GOLDBACH'S CONJECTURE IS AS FOLLOWS
-------------------------------------------------------------------------------------------------------------------

Pursuing this line of attack, let's start with an arbitrary even number n[0]:

Let
n[0] = p[0] + q[0]
and
n[1] = q[0] - p[0]
where p[0] and q[0] are distinct odd prime numbers, p[0] < q[0].

Now let
n[1] = p[1] + q[1]
and
n[2] = q[1] - p[1]
where p[1] and q[1] are distinct odd prime numbers, p[1] < q[1].

We continue in this manner until we arrive at
n[m] = p[m] + q[m]
and
n[m+1] = q[m] - p[m]
where p[m] and q[m] are not necessarily distinct odd prime numbers, p[m] <= q[m], and n[m+1] = 0, 2, or 4.

We may reconstitute n[0] as
n[0] = 2*p[0] + 2*p[1] + 2*p[2] + ... + 2*p[m] + (0 or 2 or 4)

Therefore n[0]/2 is sum of a finite series of prime numbers, plus arbitrarily 0, 1, or 2.

Since 2 itself is prime, we can say
n[0]/2 = p[0] + p[1] + p[2] + ... + p[m] + (0 or 1)

The question now is:
Does a number n[0]/2 exist that cannot be represented as the sum of a finite series of primes, plus 0 or 1?

Because we know that twin primes exist (two primes whose difference is 2),
and
because we may add 0, 1 arbitrarily,
and
because every interval (p, 2*p) contains at least one prime number,

we must conclude that the answer is NO.

Therefore Goldbach's Conjecture has been proved TRUE

fused_bulb

see...either i am crazy or u all are crazy....

British publisher Tony Faber offered a $1,000,000 prize for a proof

of the conjecture in 2000, if a proof was submitted before April

2002. The prize was never claimed....

i dont need to say more...

debabratanag99

The goldbach's conjecture is now a proved one........!!!!!!!!!!!!!!!!!!!

Only that people are validifying the proofs........!!!!!!!!

fused_bulb

u can check wikipedia...it isnt ...

debabratanag99

That is because the proofs are under scrutiny.........and once scrutinised properly then it is solved.......!!!

arvind.b

ppl.

please do not waste time trying to do stuff with the goldbach conjecture. once you get into iit you will have enough time :)

rht_ald

hey!!!!!!!!!

see ........

since every prime no. except 2 are odd.............

therefore if we add any 2 prime no. except 2(i.e. odd no.+oddno.)

the result is always even..........(since odd no. + odd no.=even no.)

hence proved............

rate me if u like ma ans..........

ayush007

we know one thing for sure...........2 is the only even no.

For prime numbers a,b,c

. . . . . a - b = (a + c) - (b + c) . . . . . even integer . . . . . . . . . . (1)

and thus, generally,

. . . . . a - b = 2k . . . . . . . . . . . . . k = integer . . . . . . . . .. . . (2)

and since a + b is an even number

. . . . . a + b = 2n . . . . . . . . . . . . n = integer>2 . . . . . . . . . . . . . (3)

Now, using (2) and (3) results in

. . . . . a = n + k . . . and . . . b = n - k . . . n>k . . . . . . . . . . . . . (4)

I can prove Goldbach's Conjecture, with reference to equations (2) ,(3)&(4).this proof of mine is just a simple 'by contradiction proof'-----suppose that not every sum of two primes a and b is an even number 2n>4(but this isn't possible as...2& 4 both are multiples of 2). However, since the number of twin primes is unlimited, their sum is an even number 2n unlimited and every sum of primes is an even number.

its quite, as simple as that ....

BUT, remember.....many times...the mathematical proffesors may also find flaws in what one persumes to be perfectly correct........
they may call it a" not universally accepted solution"...so at this stage this is the most we can do.......

keep smiling............

fused_bulb

matlab had ho gayi....

even an exper is telling u not 2 mess around with this problem

n still , evrybdy is trying to prove themselves geniuses...

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