idea of proof Legendre Conjecture

Question

I was wondering, why is Landau first problem about Legendre conjecture remain unproven?

Do we don't have complete list of prime less than 1.7 \times 10^{14} ?

Since we know that $\Delta p=p_{n+1}-p_n\leq 1.3\times 10^7$ based on Improving Zhang's prime gap
isn't supposedly proof the conjecture to $[\, 1.7 \times 10^{14} \, ,\, \infty \,)$

am i wrong? or did i incidentally proof the conjecture?

Answer 1

What Zhang, Maynard, and other mathematicians proved is that there is a constant $M$ such that the inequality $p_{n+1}-p_n\le M$ is true infinitely times. However, it is trivial that you can find infinitely many primes lying in between consecutive square, so results on small gaps between primes won't be helpful.

It is known that the prime numbers and the nontrivial zeros $\rho$ of $\zeta(s)$ are related by: $$ \vartheta(x)=\sum_{p\le x}\log p=x-\sum_\rho{x^\rho\over\rho}+\text{error terms}, $$ so estimates of zeros with $\Re(\rho)>\frac12$ is often applied to this formula to obtain results like the following:

There exists $0<\theta<1$ such that there is always a prime in $[x-x^\theta,x]$.

If we can prove the above statement with some $\theta\le\frac12$, then Legendre conjecture is proven, and the current record is $\theta=0.525$. If we assume Riemann hypothesis or Lindelöf hypothesis (which is implied by RH) then we can set $\theta=\frac12+\varepsilon$, but this is still far from proving the conjecture.

However, if we relax the condition of primes to almost primes, then we have some other approximations to Legendre's conjecture:

In 1975, J.-R. Chen proved that for large integer $x$, there always exists a number in the interval $n\in[x-x^{0.5},x]$ such that $n$ is a product of at most 2 primes. This translates to the following:

For large $N$, there always exists prime or product of exactly 2 primes between $N^2$ and $(N+1)^2$.