Recently one of my friend came up with something which he claimed to be a proof of the famous Legendre Conjecture. Let me brief his argument.
Statement of The Conjecture
There exists at least one prime between two consecutive squares.
Proof
If there existed at least one prime between two consecutive squares and if $n^2$ would be the $k$-th composite, denoted by $C$ $(k)$ then Legendre's Conjecture says that $(n+1)^2$ is impossible to become $C$ $(k+2n+1)$ . So the goal now is to find $k$ 's so that
$C$ $(k+2n+1)$ $-$ $C$ $(k)$ $>$ $2n+1$
Now from find the bound of $n$ and then maintaining the bound find all possible $n$ (his claim was this will be all integers because of the bound, and I think here is the mistake but still it is my guess, still no rigorous proof of this assertion is in my hand) and when you find those $n$ the conjecture is proved.
Let me tell now why I think that there must be something wrong in the 'proof'. Right now I can think of only two reasons.
Though it will not be logical to say but I think that the problem which has baffled so many mathematicians for so many years, is unlikely to be solved in this way.
I doubt about the correctness of the method of obtaining the bound.
Your proof of the Legendre conjecture starts with "if the Legendre conjecture is true". Therefore, it is of the form "if A, then something something something then A", which can be shortened into "if A, then A".