Hi Richard and all:

Thanks for your question.

What I mean by prime divisors is definitely any and all divisors

< sqrt x, so the 15 and 1 would qualify. I didn't mention them

because I wanted to give an example, not necessarily a comprehensive

example, but I should have included all of them. If you look at the

definition of LE you will see that 1 is eliminated from the list of

primary divisors used in the analysis.

Let me walk you through the part of the solution that has to do

with primary divisors. Most of this information is in the first

example (N=12), but perhaps a bit criptically.

Given N =12, how many primes, if any, are between 12^2 and 13^2?

First we calculate the number of possible primes (KE) between 144

and 169. We know that the only possible primes are the odd numbers,

being 145,147,149,151,153,155,157,159,161,163,165,and 167. These

numbers define the KE set. Note that the set has 12 values, so KE

=12 (KE itself is not a set, just the number of values in the KE

set). We could use some intelligence and say that we know some of

thes numbers are not prime (like 155 and 165, each divisible by 5)

but it will be difficult to keep track of all that, so we will just

leave them in as possible candidates for this analysis.

Next we must calculate the number of primary divisors that may be

legitimately used to disprove that a number in the KE set is not

prime, i.e. not divisible by anything other than itself or 1. We

first calculate the primary divisors,i.e. odd numbers less than N

(for N even), and in this example they are 1,3,5,7,9,11. Then we

calculate how many of thes values we can use as divisors to negate

primality of numbers in the KE set. The result I defined as LE, and

in this example are 3,5,7,9, and 11, all the primary divisor values

except 1.

Therefore, LE = 5 (3,5,7,9,11 constitute 5 values). Note that the

definition of LE in my post does exclude 1 so everything is OK so

far.

This takes us to a point where we have the number of possible

primes defined, as well as the values that can be used as their

divisors. Rather than explaining further, I'll wait for questions so

I can address specifically what is not clear. Don't forget to look

at the examples (especially N=12) to aid in following my thoughts.

PS I don't think I have yet mastered responding to posts. Do you

just type in the original title with a RE: at the start?

Regards, Bill