Average gap between coprimes

[J.Ashutosh]

?Namaste.
I have a proving problem.
Given: For every prime p, the average gap between the numbers which are coprime to all the primes till p is at the maximum 2p.
( Above statement is to be assumed true irrespective of fact).
To prove: There exists a prime p, for which the smallest number coprime to all the primes till p is at a gap less than 2p.(means below 3p,i.e.p+ 2p)
Taking the above assumption.
Thanks

 

[Deleted member 4993]

?Namaste.
I have a proving problem.
Given: For every prime p, the average gap between the numbers which are coprime to all the primes till p is at the maximum 2p.
( Above statement is to be assumed true irrespective of fact).
To prove: There exists a prime p, for which the smallest number coprime to all the primes till p is at a gap less than 2p.(means below 3p,i.e.p+ 2p)
Taking the above assumption.
Thanks

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520​

Please share your work/thoughts about this problem.

 

[J.Ashutosh]

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520​

Please share your work/thoughts about this problem.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520​

Please share your work/thoughts about this problem.

Actually, I started proving by contradiction, but I am not able to proceed forward or just initiate the proof.
It will be of great help if I just get the overall proof.
Thank you

 

[Steven G]

No, no one here will give you the proof. This is a math help forum.

Show us what you have done so far and you will be told where you made a mistake and/or a hint to what to try next.

 

[apple2357]

Why do you want to just see a proof that someone else has done? I'd be interested in why you are interested in this problem and where it came from? It sounds like a nice problem to think about.

 

[J.Ashutosh]

To be honest, I was working on a problem and there I arrived at the result, given in the above proving statement. But I am stuck on this seemingly easy problem, of which I get a clear intuition. And the intuition is like
If we have n boxes with some total amount of objects in them. If the average number of objects in a box is x, then at least one box must contain at the most ( and also at the least) x objects.
If anyone ,with a fresh perspective of the problem, could provide me help on this one , it would be of a great significance.