Back in the spring of 2002 I discovered the following way to count
prime numbers:

dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,
sqrt(y-1))],

S(x,1) = 0, S(x,y) is the sum of dS from dS(x,2) to dS(x,y),

p(x, y) = floor(x) - S(x, y) - 1,

and p(x,sqrt(x)) gives the count of primes where, of course, only
positive integers are being used, so the sqrt() operator returns the
largest positive integer that squares to give the argument or a number
less than it.

Back in the spring of 2002 I discovered the following way to count
prime numbers:

dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,
sqrt(y-1))],

S(x,1) = 0, S(x,y) is the sum of dS from dS(x,2) to dS(x,y),

p(x, y) = floor(x) - S(x, y) - 1,

and p(x,sqrt(x)) gives the count of primes where, of course, only
positive integers are being used, so the sqrt() operator returns the
largest positive integer that squares to give the argument or a number
less than it.

Back in the spring of 2002 I discovered the following way to count
prime numbers:

dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,
sqrt(y-1))],

S(x,1) = 0, S(x,y) is the sum of dS from dS(x,2) to dS(x,y),

p(x, y) = floor(x) - S(x, y) - 1,

and p(x,sqrt(x)) gives the count of primes where, of course, only
positive integers are being used, so the sqrt() operator returns the
largest positive integer that squares to give the argument or a number
less than it.

Now consider values for dS(x,y) from the calculation of p(100,10):

dS(100,2)=49
dS(100,3)=16
dS(100,4)=0
dS(100,5)=6
dS(100,6)=0
dS(100,7)=3
dS(100,=0
dS(100,9)=0
dS(100,10)=0

The sum of all those values gives S(100,10)=74, which is the count of
composites, 74 subtracts from 100 to give 26, and subtracting 1 from
that gives you p(100,10) = 25, and 25 is the count of primes up to
100.

Now then dS(x,y) is the count of composites that have y as a factor
that do NOT have a prime number less than y as a factor up to and
including x, so it makes sense that unless y is a prime dS(x,y) is 0
because, for instance, if y=4, then any number that has 4 as a factor
also has 2 as a factor, so there aren't any composites that have 4 as
a factor that do not have a prime number less than 4 as a factor as,
of course, 2 is a factor of 4.

Practically that means that dS(x,y) is 0 if y is composite, and
non-zero if y is prime and less than or equal to sqrt(x) as can be
seen with the values given above.

Now then the non-zero values are easy to understand, like
dS(100,2)=49, means that there are 49 numbers up to and including 100
that are even, excluding 2 itself, as its prime. Then dS(100,3)=16
tells you that there are 16 numbers that are composite with 3 as a
factor that are NOT even.

Next you have dS(100,4)=0, as there aren't any numbers that have 4 as
a factor that do not have a smaller prime as a factor, in this case 2,
as 4 has 2 as a factor.

Next there's dS(100,5)=6, telling you that there are 6 composites with
5 as a factor that do not have 2 or 3 as a factor.

Notice the drop of non-zero dS(x,y) values is like an exponential
drop, while the spacing between values is, of course, the spacing
between prime numbers.

And that explains:

dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,
sqrt(y-1))]

More information can be found at my blog:

http://mathforprofit.blogspot.com/


James Harris