Spot the error!

[Lior]

Hello Guys,
A few days ago, I saw a letter the mathematics department of some university which I don't remember its name received claiming for a "proof" for the famous 250-years unproved Goldbach's Conjecture.
Let me just me remind you that the conjecture claims (After Euler's reformulation):

Every even integer bigger than 3 is a sum (in perhaps more than one way) of two primes (not necessarily distinct).

OK, Concentrated? if not, warm-up your brain a little. The "proof" is pretty simple, no high mathematics knowledge required.
So here comes the "proof", just try to spot the error. (I found it after a while)


A conjecture clearly equivalent to Goldbach's is:
Conjecture A: Every even integer greater than 5 is a sum of three primes.
Another conjecture is:
Conjecture B: Every even integer greater than 5 is a sum of two primes (both odd).

Theorem AB: Conjecture A and B are equivalent, either both true of both false.
Proof: Choose any even integer 2n > 5. If conjecture A is true then 2n+2, another even integer greater than 5, must be a sum of three primes;
they cannot all be odd, so one of them is 2, and this implies that 2n is a sum of two odd primes, implying conjecture B.
On the other hand, if conjecture B is true then 2n-2 , is either 4 or an even integer greater than 5, must be a sum of two primes;
This implies that 2n is a sum of three primes, which confirms Conjecture A. Thus, Conjectures A and B are either both true or else both false.

Theorem C: If any even integer that is a sum of three primes is a sum of two primes too, then Goldbach's conjecture is true.
Proof: If Goldbach's conjecture is false, there must be an even integer 2n bigger than 5 which is first to violate the conjecture and not to equal the sum of two primes.
Therefore, 2n-2 is a sum of two primes. This implies 2n is a sum of three primes.
This can't happen if any even integer that is a sum of three primes is a sum of two primes too.
Therefore, if any even integer that is a sum of three primes is a sum of two primes too, Goldbach's conjecture is true.

Theorem G: Any even integer that is a sum of three primes is a sum of two primes too.
Proof: Here is a proof by contradiction. If theorem G were false, there would be an integer 2n which was the sum of three primes but not the sum of two primes.
Such an integer would violate Conjecture B without violating Conjecture A, contradicting the equivalence of those conjectures proved in theorem AB.
Therefore, theorem G can't be false; it must be true.

Combining theorems G and C proves Goldbach's conjecture.

Looks logical? think twice...

[bugzpodder]

Conjecture A and B are equivalent, either both true of both false.

what happens when they are both false?

[Lior]

You're on the right direction, develop it...

[Lior]

Welp? no one wanna give a try?

[bugzpodder]

that won't happen. y don't u give out the answer?

[Lior]

what's the point? it's not an article, it's a riddle.