Quote:
Originally Posted by MathDoggy
Introduction: The twin prime conjecture is a mathematical hypothesis which states that there exists infinitely many prime numbers that differ by 2.
Proof by direct method:
Let s be the infinite sum of the first n twin prime numbers
Let z be the infinite sum of the first n natural numbers
Let us assume that there exists a finite amount of twin prime numbers, then by the comparison criterion we will check whether the infinite sum of z diverges or converges because if z diverges then by implication s will also diverge.
Now we will prove the divergence of z:
z=1+2+3+4+5+6+7..
Now let us take the largest power of 2 which is greater or equal to K, where K is an element of z distinct to 2.
Now we have,
z=0+1+2+2+2+2+3+3...
Obviously this infinite sum diverges then we can conclude that s also diverges therefore there does not exist a finite amount of twin prime numbers.
Q.E.D
(I have used the same method to prove this conjecture as the Goldbach conjecture)
If you want to see the other proof, you can take a look in the Algebraic Number Theory section.

Can somebody point out what parts of the proof are wrong