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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