Like my way to count prime numbers which is SO CLOSE to what was known but is more compact because of innovative approach.

Will give the sieve form as is smallest to show, and actually is MUCH prettier with the summation sign but not so sure on how to do that with Blogger or if I can. Interested readers can just write it out to see it that way.

With integers ONLY, where p

_{j}is the jth prime:

P(x,n) = x - 1 - sum for j=1 to n of {P(x/p

_{j},j-1) - (j-1)}

The P(x,n) function will count prime numbers up to and including x, if n equals the count of primes up to sqrt(x), and if as you iterate you never let n be greater than the count of primes up to and including sqrt(x), if the function receives an n greater than that value it just needs to reset it to that count.

That's mine. I found it.

For example P(100,4) = 25.

(Wish to pursue more? I use labels on most posts, and beneath this one should see the prime counting label. Click on it for more related posts!)

My way of doing research was very disruptive as I wasn't so excited about mathematical errors, as much as idea generation, and tossing of erroneous approaches rapidly. Where I used the web for feedback to help figure out when things were wrong. And now finally with the web can distribute perfect ideas which withstood all challenges.

As mathematics is awesome in a fascinating way: correct ideas can't be broken, cannot fail, cannot lose against challenge. So why not hit them with everything you've got?

The math doesn't care.

James Harris