 # Number Theory: Do most composite numbers have a large prime factor?

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Do most composite numbers have a large prime factor?

First, I’ll define what I mean by a “large” prime factor.  Let N be a number.  If a prime factor of N is greater than the square root of N, then that factor is a large prime factor of N.

As an example, 11 is a large prime factor of 22, because 11 is greater than the square root of 22, and so 22 has a large prime factor

On the other hand, 3 is not a large prime factor of 12 because 3 is less than the square root of 12, and so 12 does not have a large prime factor.

Below is a list of composite numbers with large prime factors:

6, 10, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, …

It seems that, as numbers increase, a greater and greater percentage of them have large prime factors.  I say that that seems to be true, because I have sampled some groups of big numbers, and most of them had large prime factors.  Of course, that isn’t proof and as far as I know it could also be wrong.

If we check all of the numbers up to 330, the majority of counting numbers are composite numbers with large prime factors.

If I understand it correctly, then what I’m asking about is similar to the question answered by the Prime Number Theorem.  According to the Prime Number Theorem, for a very large number N, the probability that a random integer not greater than N is prime is equal to 1/log(N).

Because the prime numbers are distributed in this way, and 1/log(N) can be arbitrarily close to zero, the composite numbers can be seen as essentially the same as all integers, for very large values of N.  For very large numbers, my question is the same as asking what percentage of all integers have a large prime factor.

My question is, “For a very large number N, what is the probability that a random integer less than N has a large prime factor?”  “Is this probability greater than 0.5?”  I’m hoping there might be some kind of answer to this in the same way that the Prime Number Theorem answers the question about the distribution of prime numbers.

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