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?

 

[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.

 

[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.