IIT JEE-AIEEE 2012, Entrance Notifications, Study Tips, Engineering Preparation, Practice Test Papers, Experts Advice

DEBABRATA NAG

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
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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

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