A Generalization of Erdos’s Proof of Bertrand-Chebyshev Theorem

Bertrand’s postulate states that for every positive integer n, there is always at least one prime psuch that np2n. 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 n2 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 knp(k+1)n for all integer n>1 and k <=n ?

A positive answer for kn 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 < kn, 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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