Saturday, August 28, 2004


My prime counting formula, other prime counting

For over two years I've talked about my prime counting discovery only to face a strange apparent lack of interest from mainstream mathematicians and outright hostility from mostly sci.math posters who tend to try and claim my work is not new.

Now I'm going to explain just a bit on how my work connects with what mathematicians found on their own.

For instance one particular sci.math poster has for years now charged that my work is just a copy of something called Legendre's Formula!

It turns out that Legendre's Formula uses a partial sieve function called phi(x,a) where phi(x,a) is the count of positive integers less than or equal to x which do not have the first 'a' primes as a factor.


The historical prime counting function is pi(x), which can be confusing to some as they see "pi" in front, but it's just the count of primes up to and including x.

Still pulling info from that page on Legendre's Formula:

phi(x,pi(sqrt(x))) = pi(x) - pi(sqrt(x)) + 1

So you can solve for pi(x), and if you figure out pi(sqrt(x)) and phi(x,pi(sqrt(x))) then you have a count of prime numbers.

Notice that phi(x,a) is a partial sieve function because you actually have to know what the a_th primes are. Like if a=2, it is fed the prime numbers 2 and 3.

Now I'll show how phi(x,a) works with a simple explanatory example.

For example, up to and including 10 you have

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

and pi(sqrt(10)) = 2, and those primes are 2 and 3.

So phi(10,2) = 3, and those numbers are

1, 5, 7

as those are the ones not divisible by 2 and 3, now since

phi(x,pi(sqrt(x))) = pi(x) - pi(sqrt(x)) + 1

you can solve for pi(x) to get

pi(x) = phi(x,pi(sqrt(x))) + pi(sqrt(x)) - 1


pi(10 = 3 + 2 - 1 = 4

which is the correct answer.

Notice you add back in to count 2 and 3 themselves, and subtract 1 for 1 which is counted in phi(x,a), and the 4 primes are

2, 3, 5, 7.

Now my prime counting function is

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, p(x, y) = floor(x) - S(x, y) - 1, and S(x,y) is the sum of dS from dS(x,2) to dS(x,y)

where dS(x,y) is the count of composites up to and including x that have y as a factor which do not have any primes less than y as a factor, and the count of primes is given by p(x,sqrt(x)), so to connect with the traditional:

pi(x) = p(x,sqrt(x))

Now if you're counting primes you are just counting primes, so it's not surprising that various methods, though looking dramatically different MUST have underlying similarities because they're counting the same thing.

Now I explained my dS(x,y) function above but the S(x,y) function is the count of composites up to and including x which have the first primes up to and including y as factors.

Like S(10,3) = 5, and those composites are

4, 6, 8, 9, 10

so how do my dS(x,y) and S(x,y) relate back to Legendre's Formula?

The connecting relationship is

phi(x,a) = x - S(x,p_a) + pi(sqrt(x)) + 1

where I have to actually show the prime that is being referred to by the sieve in my S(x,y) function.

Now the physicists here can appreciate that, sure, you can have connecting formulas behind functions that ultimately do the same thing, but sci.math'ers have claimed for over two years, including recently in posts here, that the relationship "essentially" proves that my prime counting function is just Legendre's Formula.

It gets weirder though, as the phi(x,a) partial sieve function can be defined by a recurrence relationship:

phi(x,a) = phi(x, a-1) - phi(x/p_a, a-1)

which looks a lot like part of my dS(x,y) function, as consider

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

and if you have that y is a prime, like let's say p_a, then

dS(x,p_a) = (p(x/p_a, p_a-1) - p(p_a-1, sqrt(p_a-1)))

where since

p(p_a-1, sqrt(p_a-1)) = pi(p_a-1)

so I have that

dS(x,p_a) = p(x/p_a, p_a-1) - pi(p_a-1)

and multiplying both sides by -1 and reordering a bit I finally have

-dS(x,p_a) = pi(p_a-1) - p(x/p_a, p_a-1)

and pulling down the recurrence relationship above so it's easier to directly compare

phi(x,a) = phi(x, a-1) - phi(x/p_a, a-1)

and yes they look similar!

One poster in particular has, for over two years now, repeated over and over again that my prime counting function is just Legendre's Formula based on just that similarity, but look closely, and it seems actually a bit odd that someone could make the case for over two years that those are exactly the same, and get away with it, while another sci.math'er has been more creative in claiming they are "essentially" the same.

You can imagine my consternation as a discoverer with having done the work to find my prime counting function only to see mathematicians not acting as expected while sci.math'ers would say the strangest things.

So, the short answer is, yes I can relate my work to a certain extent to what mathematicians already had, though notice I have to constrain my functions just a bit for them to connect with the sieve function.

That's easy enough though as instead of "y" I just put in "p_k" where k means the k_th prime.

But try to go the other way from what mathematicians have to my work.

The short of it is that there's no rational reason to accept the charge that my work is the same as Legendre's Formula but by now I'm sure many of you realize that the issue here isn't rationality.

It's about social stuff, academic politics, and a society of people—mathematicians—who clearly feel that they can make up their own rules.

Now I've given the derivation of my prime counting function, which you can read for yourself at my blog:

It is actually a nice introduction to the concepts of counting prime numbers, and all of the formulas make sense, and you don't need to be a top mathematician to understand it!

Now sci.math'ers have made this all personal, but it's knowledge.

Sure, many of you may just accept that mathematicians have the right to hide whatever they want, but it'd be like accepting that physicists should have the right to hide physics research.

Many of you may believe that it's my fault, and that I should have kissed butt or whatever it took to get mathematicians to like me enough to acknowledge my results, but you forget, I have the knowledge.

It's the world that is being blocked from it. Not me.

And to me, knowledge is a great thing, sure in and of itself it's not power, and I think my experiences show dramatically that knowledge is NOT power, as social forces allow people to dismiss just about anything.

Social forces are power. People can get together and simply tell themselves what they wish to believe and there's little you can do about it. There's no sense worrying about it too much as that's just reality.

Knowledge is NOT power.

But it's still immensely satisfying when you're standing on top of a mountain surveying a world. Sure it might be a little lonely, but I wouldn't have it any other way.

Hey, I found my own way to count prime numbers which advanced the literature, with one of the most beautiful derivations ever.

People act wacky about it? Sure! Of course they would!!! Of course mathematicians would behave screwy!!!

Of course. You see, they didn't make the find.

I did.

There's some guy I saw on TV who is trying to get in the record books for surfing every day for some number of years…

Someone has a world record for walking backwards…

Athletes currently are pushing themselves at the Olympics and some will receive medals.

People work to get attention, and society rewards them for standout accomplishments.

Mathematicians are just cheating with my work.

They are remaking the rules where a significant individual accomplishment can be ignored if the group charged with recognizing such accomplishments doesn't feel like it.

The are cheating in a world that knows the value of acknowledging accomplishment.

They have to know what they are doing.

The facts are on my side. I have my prime counting function itself, the derivation, and several different ways to show it to be important, unique, and even beautiful, but mathematicians are cheating.

Look at the Olympics and imagine the world the mathematicians would have, where if you don't like that runner, like if she makes you angry, you can refuse to give a medal.

If that guy throwing that javelin happens to not be nice to you or just because you don't really feel like it that day, who cares if his throw is a record?

You can just ignore him.

Mathematicians have been known for a while to be kind of out of it, and sort of in their own little world, but now it's clear that part of that world is deliberately ignoring an accomplishment like mine.

The proper punishment for them is to ignore what they claim are accomplishments.

Let them taste the pain they are willing to give to others.

Put the math people in the doghouse where they belong.

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