DIrect Proof - primes and factorials (1 Viewer)

[gamja]

Is this possible to be solved using direct proof? From MANSW chal q

Also, what do you begin to consider in general with proof questions with primes or factorials in them?

 

[tywebb]

It follows from Bertrand’s Postulate: For all integers n>1 there exists a prime p such that n<p<2n.

For n≥3, 2n≤n! and therefore n<p<n!.

In any case there are some proofs here: https://math.stackexchange.com/questions/483838/for-all-n2-there-exists-p-prime-npn

 

[cossine]

Also, what do you begin to consider in general with proof questions with primes or factorials in them?

Often when we are given a question it is not clear what approach to take. As result we just need to give it go and try different approaches maybe do some research as tywebb did with math stackexchange.

 

[Eagle Mum]

But where tf did all the numbers that are meant to be in math go

The symbols represent sets of numbers with specified conditions.

 

[Eagle Mum]

It follows from Bertrand’s Postulate: For all integers n>1 there exists a prime p such that n<p<2n.

For n≥3, 2n≤n! and therefore n<p<n!.

In any case there are some proofs here: https://math.stackexchange.com/questions/483838/for-all-n2-there-exists-p-prime-npn

I guess technically, in an exam or competition, one would need to add that the postulate is now a theorem /has been proven, to show that you understand that a postulate can’t be used to prove another maths statement.

 

[tywebb]

It is a well known theorem. Tchebychev proved it in 1850. Then in 1919 Ramanujan made a shorter proof. Then in 1932 Erdős made a more elementary proof which is the one most commonly used thesedays, for example at https://en.wikipedia.org/wiki/Proof_of_Bertrand's_postulate

Here is a youtube explaining it which is bit easier to follow than the wikipedia article:

 

[Eagle Mum]

It is a well known theorem. Tchebychev proved it in 1850. Then in 1919 Ramanujan made a shorter proof. Then in 1932 Erdős made a more elementary proof which is the one most commonly used thesedays, for example at https://en.wikipedia.org/wiki/Proof_of_Bertrand's_postulate

I get that - it was more the general principle of communicating. That one should demonstrate understanding that when a proof is required, every step should be proven, so it’s unequivocal to state Bertrand‘s postulate (proven) or Bertrand’s postulate (now theorem). I’m not a maths marker (maths is just a hobby), but I am a postgrad exam marker and have peer reviewed many publications, so I guess I look out for such details.

 

[tywebb]

A reference or proof should be given if it is not well known.

But for well known theorems it is not necessary.

For example one does not need to prove Pythagoras' theorem or give a reference to a proof every time one uses it. One only needs to say "by Pythagoras' theorem, ...."

There are 2 reasons Bertrand's postulate is called that and not "Bertrand's theorem".

Firstly although he first made the postulate in 1845, he wasn't the one to prove it. He did however give some numerical evidence with some examples. It is sometimes referred to as Tchebychev's theorem because he was the first one to prove it 5 years later. Nevertheless "Bertrand's postulate" is more commonly used to give credit to Bertrand because he came up with the idea, not Tchebychev.

The second reason is to distinguish it from another theorem which is called "Bertrand's theorem", which is something completely different related to mechanics.