Conjectures by Legendre and Brocard made stronger.

  • Thread starter Rudy Toody
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  • #1
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Main Question or Discussion Point

If this series https://www.physicsforums.com/showthread.php?t=485665 is proved to be infinite, then proofs of these two conjectures can be done as simple corollaries.

Legendre's Conjecture states that for every $n\ge 1,$ there is always at least one prime \textit{p} such that $n^2 < p < (n+1)^2$.

Our stronger conjecture states that for every $n\ge 1,$ there are always at least \textbf{two} primes \textit{p} such that $n^2 < p_{m},p_{m+1} < (n+1)^2$.

Brocard's Conjecture states that for every $n\ge 2,$ the inequality $\pi((p_{n+1})^2)-\pi((p_n)^2) \ge 4$ holds where $\pi(n)$ is the prime counting function.

Our stronger conjecture states that for every $n\ge 2,$ the inequality $\pi((p_{n+1})^2)-\pi((p_n)^2) \ge 2(p_{n+1}-p_n)$ holds where $\pi(n)$ is the prime counting function.

Sorry, I couldn't get the [tex] stuff to work.
 

Answers and Replies

  • #2
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Is conjecturing a stronger conjecture really progress? It's not like proving a stronger theorum then has already been proved.
 
  • #3
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Is conjecturing a stronger conjecture really progress? It's not like proving a stronger theorum then has already been proved.
The important theorem would be the one that proves the function in the first link. I'm trying to show how important that function is. It is magical. It can be used for additional proofs, too.
 

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