Harristotelian Logic

It turns out that my prime counting research is a crusher, which is why posters have to lie differently about it, as for instance, they can't claim it's wrong!

With natural numbers x and n, where p_i is the i_th prime:

P(x,n) = x - 1 - sum for i=1 to n of {(P(x/p_i,i-1) - (i-1))}

where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

Here rather than give LaTex I've copied a bit from a sci.math poster.

That is just a neat and tidy prime counting function which differs from Legendre's in a key way, as with Legendre's you use exclusion-inclusion where you get counts of composites using the primes.

So to count the primes up to and including x you start with floor(x/2) which gives you the count of evens, and next you use floor(x/3) as you go up by prime, BUT now you have a problem!!!

Some of the composites that have 3 as a factors are even, so you have over-counted and Legendre's corrects by now adding back in floor(x/6).

WHAT I did in contrast was say, hey, find out all the evens, but now when you look at numbers that have 3 as a factor, don't bother counting those that are even! So my prime counting function is different in a simple way: it counts composites by prime excluding the counts of composites already found with a lesser prime.

May not seem like a big deal but that leads to a short prime counting function in its sieve form as I showed above.

But you can easily speed it up as well!!!

With natural numbers x and n, where p_i is the i_th prime:

P(x,n) = x - floor(x/2) - 1 - sum for i=2 to n of {(P(x/p_i,i-1) - (i-1))}

where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

Now it's over TWICE as fast! Just like that.

That cannot be done with Legendre's method. It just can't be written that succinctly or sped up as easily and YES you can keep going!!!

You can keep speeding it up!

Now notice, I just quickly gave you key facts that make my prime counting function different in crucial ways, and it is neater and more concise as well.

Posters have special problems in counter-acting the truth of my research here, so usually they simply go to making things up!!!

My suggestion to those of you who care even a little bit about primes it to just code what I have above, and speed it up a bit. Then go and look at the clunky stuff that mathematicians continue to obstinately use, as if they don't have a working neuron in their heads.

Remember, my prime counting function differs in that I went to the common-sense idea of counting at each prime ONLY those composites that have that prime as a factor but NO LESSER PRIMES as a factor.

Here's a tidbit to play with as floor((x-3)/6) gives the count of composites that have 3 as a factor but do NOT have 2 as a factor, for instance with x=12, floor((12-3)/6) = 1.

So what is that one?

It's 9 of course. Um, but what about 3?

Well 3 wasn't counted because it's not composite!!!

And of course 6 and 12 were left off because they are even.

Don't think the math can be that smart with such a simple expression?

Now try x=15, as now you get floor((15-3)/6) = 2.

It just counted 15 because it has 3 as a factor and is NOT even!!!

It's just a smarter way to count primes and it leads to a compact prime counting function in its sieve form.

Now then, when you have a better idea that a LOT of people are fighting you have to wonder what might be the deep secret to why.

Answer here is, the sieve form is boring. It turns out that you can go to a more complex fully mathematicized form, which uses what's called a partial difference equation. You keep pulling that thread and you get to a partial differential equation.

You work it all out logically and you have a strong implication that the Riemann Hypothesis is false.

As a group MOST mathematicians though say the Riemann Hypothesis is true!!!

By ignoring my research they can hold on to the conclusion democratically held.