Bertrand’s postulate states that for every positive integer n, there is always at least one prime psuch that n < p < 2n. This was proved by Chebyshev in 1850, Ramanujan in 1919 and Erdos in 1932.

Legendre’s conjecture states that there is a prime number between n^{2} and (n+1)^{2} for every positive integer n. It is one of the four Landau’s problems, considered as four basic problems about prime numbers. The other three problems are

- Goldbach’s conjecture : Every even integer n > 2 can be written as the sum of two primes ?
- Twin prime conjecture : There are infinitely many primes p such that p+2 is prime ?
- Are there infinitely many primes p such that p-1 is a perfect square ?

All these problems are open till date !! Lets look at the following generalization of the Bertrand’s postulate :

Does there exist a prime number p, such that kn < p < (k+1)n for all integer n>1 and k <=n ?

A positive answer for k = n would prove Legendre’s conjecture. Recently I generalized Erdos’s Proof of Bertrand-Chebyshev’s Theorem and proved the following theorem :

Theorem : For any integer 1 < k < n, there exists a number N(k) such that for all n >=N(k), there is at least one prime between kn and (k+1)n.

Like Erdos’s Proof, my generalization uses elementary combinatorial techniques without appealing to the prime number theorem. An initial draft is available on my homepage.

I have the following question :

Are there infinitely many primes p such that p+k is prime ?

Is the answer known for any fixed *k > 2* ? What if *k* is allowed to depend on *p* ? If you know any papers addressing such questions, please leave a comment.

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Do I misunderstand or Did you prove that there s always a prime between

an -(a+1)n when n>=a ? if that is proved also legendre’s conjectured will have been proved . I also noticed that generalization and wondered if it was asked before, so I learned that the answer is yes.

I believe that can be proved using dirichlet theorem : if p is prime there exists infinitely primes in the form p+k where k is integer.

show an opposite example of the conjecture below:

There can be such a value n<(a+1)^2 , so that there can always be a+1 primes between

a.n -(a+1).n

for example between 2n-3n ,for n=8 < 3^2 , there are 3 primes : 17,19,23

for n= 14 < 4^2 , there are 4 primes between 3.12 – 4.12 which are 37,41,43,47

does this work for all n< (a+1)^2 ?