14307Primes and Squares1- clarifications
- Dec 31, 2003Hi 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
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
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?