Question

In: Advanced Math

If n>=2, prove the number of prime factors of n is less than 2ln n.

Expert Solution

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.

Colby Messinger answered 2 years ago

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