The maximum distance from any given number n to the next prime is less than twice the square root of n.
Is this statement equivalent to Legendre's conjecture? Is this statement worded correctly? If this statement were to be proven, would Legendre's conjecture be proven?
Is there another way to express Legendre's conjecture, a way which states that the upper bound on the prime gap above any perfect square is related to the square root of that perfect square, such that the statement is equivalent to Legendre's conjecture?
Taking into consideration the fact that 3 is the only prime of the form n^2-1, can we restate Legendre's conjecture to say the following?:
There will always be a prime in the interval between n^2 and n^2+2n.
Note that for n=1, 2 is between 1 and 3.