- #1
cragar
- 2,552
- 3
If K is a prime is there a prime between k and 2k.
Obviously this is a weaker version of a prime between n and 2n that was proved by
Erdos and Chebyshev.
Let's assume that their isn't a prime between k and 2k.
This would imply that all the numbers between k and 2k would have to be constructed
from primes smaller than K. When I say constructed I mean their prime factorization.
so there would have to be some product of primes that was in between
k and 2k , [itex] k<P_1P_2...P_n<2k [/itex] with [itex] P_n<K [/itex]
ok so as soon as we had one number in between k and 2k then the smallest prime that
we could multiply it by is 2, but 2 times this product of primes would be bigger than 2k
so we would not have constructed all the numbers between k and 2k.
Actually I thought I could arrive at a contradiction but I lost my train of thought.
any help would be much appreciated.
Obviously this is a weaker version of a prime between n and 2n that was proved by
Erdos and Chebyshev.
Let's assume that their isn't a prime between k and 2k.
This would imply that all the numbers between k and 2k would have to be constructed
from primes smaller than K. When I say constructed I mean their prime factorization.
so there would have to be some product of primes that was in between
k and 2k , [itex] k<P_1P_2...P_n<2k [/itex] with [itex] P_n<K [/itex]
ok so as soon as we had one number in between k and 2k then the smallest prime that
we could multiply it by is 2, but 2 times this product of primes would be bigger than 2k
so we would not have constructed all the numbers between k and 2k.
Actually I thought I could arrive at a contradiction but I lost my train of thought.
any help would be much appreciated.