In: Advanced Math
If n>=2, prove the number of prime factors of n is less than
2ln n.
Proof by induction.
For , number of prime
factors of 2 is
, hence
the statement is true for n=2.
Assume the statement is true for all i.e. for all
, number of
prime factors of m is less than
.
To prove the statement we need to show number of prime factors
of n+1 is less than
.
Suppose is
prime then note that of prime factors of n+1 is 1, which is less
than
.
So assume is
composite. Let
be a prime factor of
. Then
note that
, hence number of prime factore of
is less than
.
Note that, number of prime factors of = (number of
prime factors of
)
----------------(*)
Now note that for any prime ,
i.e.
.
Thus from (*) we get,
.
Hence we proved the number of prime factors of n+1 is less than
Hence done by induction.