## After Reimann

### After Reimann

The golden prize in number theory is to get a function P(n)=p(n)

Were P of n gives us the nth prime number.

Here is a sketch of how we may do it.

We know that n ranges over the natural numbers.

The natural numbers do have a simple function to obtain the nth natural number..simply N(n)=n

We also know that by the fundamental theorem of arithmetic that when we multiply out all the primes we gt the natural numbers.

So in essence this is what needs to be done.

P(n) must b able to morph into N(n).

So

1. we know how to get from P to N (multiply all primes)

and

2. we know what N is.

the algorithm

A.start with the natural numbers described by N

B. remove all composites made by 2 prime numbers.

C. remove all composites made by 3 prime numbers.

D. ...........

E. At each stage we generate a new sequence and function to describe it , the natural numbers with gaps.

F. at each stage we approach P, until at infinity we are left with only prime numbers and their description (function).

I hope you see that there is a hiden function besides N and P.

The function that generates accepts at first another function N(n) as M (for morph) ...M(1) = N(n) ...and M(2) ok that 2 means we now remove composites made of 2 prime numbers..so M(2)=M(1)Prime...and M(3)=M(2)prime. and the limit of this function of functions gives us P(n).

Tadaaaa!

Were P of n gives us the nth prime number.

Here is a sketch of how we may do it.

We know that n ranges over the natural numbers.

The natural numbers do have a simple function to obtain the nth natural number..simply N(n)=n

We also know that by the fundamental theorem of arithmetic that when we multiply out all the primes we gt the natural numbers.

So in essence this is what needs to be done.

P(n) must b able to morph into N(n).

So

1. we know how to get from P to N (multiply all primes)

and

2. we know what N is.

the algorithm

A.start with the natural numbers described by N

B. remove all composites made by 2 prime numbers.

C. remove all composites made by 3 prime numbers.

D. ...........

E. At each stage we generate a new sequence and function to describe it , the natural numbers with gaps.

F. at each stage we approach P, until at infinity we are left with only prime numbers and their description (function).

I hope you see that there is a hiden function besides N and P.

The function that generates accepts at first another function N(n) as M (for morph) ...M(1) = N(n) ...and M(2) ok that 2 means we now remove composites made of 2 prime numbers..so M(2)=M(1)Prime...and M(3)=M(2)prime. and the limit of this function of functions gives us P(n).

Tadaaaa!

### Re: After Reimann

Not bad. The problem is that although at each stage we have some function that describes the construction up to that step, you haven't said how the sequence of functions determines the ultimate function. In other words after stage n we have function f_n, but you haven't said how to get your ultimate function from f_1, f_2, f_3, ...Moyo wrote:

A.start with the natural numbers described by N

B. remove all composites made by 2 prime numbers.

C. remove all composites made by 3 prime numbers.

D. ...........

E. At each stage we generate a new sequence and function to describe it , the natural numbers with gaps.

F. at each stage we approach P, until at infinity we are left with only prime numbers and their description (function).

Of course you are absolutely correct that your procedure DOES output the n-th prime. What you've described is the Sieve of Erotosthenes. https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

However what you DON'T have is any formula for the resulting function. It's true that you do have a formula for the partial function at each stage n; for example you could just use a Lagrange interpolation polynomial. https://en.wikipedia.org/wiki/Lagrange_polynomial

But the polynomials change wildly for each n, and don't converge in any way to an "ultimate" sieve polynomial.

Last edited by wtf on Thu Oct 15, 2015 1:54 am, edited 1 time in total.

### Re: After Reimann

Get the limit of M(n).

M(n) is the mother function that generates the child functions which at one end are N(n) and at the other P(n).

M(n) is the mother function that generates the child functions which at one end are N(n) and at the other P(n).

### Re: After Reimann

The partial functions don't converge in any way I can think of. You need to say exactly how to determine the limit. If you could do that, you'd have something.Moyo wrote:Get the limit of M(n).

M(n) is the mother function that generates the child functions which at one end are N(n) and at the other P(n).

### Re: After Reimann

I'm going to use a very technical term...the functions become more "primey". lolwtf wrote:The partial functions don't converge in any way I can think of. You need to say exactly how to determine the limit. If you could do that, you'd have something.Moyo wrote:Get the limit of M(n).

M(n) is the mother function that generates the child functions which at one end are N(n) and at the other P(n).

longer explanation just now

### Re: After Reimann

Lets plot the prime numbers evenly on a line and call it axis A.

Then get another axis B and put it perpendicular to form a plane. On that plane exists all the numbers made by multiplying two prime numbers. If we were to draw a line or a circle on this plane the functional description would be very different from the normal real number cartesian product. We could start by describing points o the plane then move on perhaps by finding out how all the different theorms we have on prime numbers can be expressed as geometric figures on this plane. Once we have expressed these we would now , because of symmetry, be able to define the plane or planes by extension (adding more and more prime number planes to create composites made by n_prime numbers).

Each plane represents a particular M(x).

Then get another axis B and put it perpendicular to form a plane. On that plane exists all the numbers made by multiplying two prime numbers. If we were to draw a line or a circle on this plane the functional description would be very different from the normal real number cartesian product. We could start by describing points o the plane then move on perhaps by finding out how all the different theorms we have on prime numbers can be expressed as geometric figures on this plane. Once we have expressed these we would now , because of symmetry, be able to define the plane or planes by extension (adding more and more prime number planes to create composites made by n_prime numbers).

Each plane represents a particular M(x).

### Re: After Reimann

Can you give an example to say what you mean? So on the A axis we have the prime 7, say. Then what's on axis B? Is axis B another copy of the primes, so you have a point (5,7), say, which represents 35? I'm not following what's on axis B.Moyo wrote:Lets plot the prime numbers evenly on a line and call it axis A.

Then get another axis B and put it perpendicular to form a plane. On that plane exists all the numbers made by multiplying two prime numbers.

Where are the products of 3 primes? On a different plane? A different axis?

And still, even if you have one plane (or axis) for each prime, you don't say how to take some kind of limit and get your ultimate function.

### Re: After Reimann

Each axis contains a progression of primes, so the A axis is (2,3,5,7,...) and the B axis is also (2,3,5,7...) and so are all the rest. If we form a cartesian plane using A and B then on the plane we can only have composites made of 2 prime numbers...e.g. 2X3 = 6 ; 5X11 = 55...etc...wtf wrote:Can you give an example to say what you mean? So on the A axis we have the prime 7, say. Then what's on axis B? Is axis B another copy of the primes, so you have a point (5,7), say, which represents 35? I'm not following what's on axis B.Moyo wrote:Lets plot the prime numbers evenly on a line and call it axis A.

Then get another axis B and put it perpendicular to form a plane. On that plane exists all the numbers made by multiplying two prime numbers.

Where are the products of 3 primes? On a different plane? A different axis?

And still, even if you have one plane (or axis) for each prime, you don't say how to take some kind of limit and get your ultimate function.

To get the composites made of 3 prime number we need to add yet another plane perpendicular to the other two C =(2,3,5,7...) on this 3D "surface we will find numbers like 110 = 2X5X11 and 231 = 3X7X11...etc...

If we add an infinite number of these axis we will on this infinityD plane , all the natural numbers.

Each plane would be defined by M..so M(2) = N(n)prime as i wrote in the OP is associatd with the plane of this cartesian of primes structure with 2 Axis.

Once we have this , all the theorems we have on prime numbers can be represented on this structure as geometric structures..if we have enough structures we can "work out" perhaps using group theory of some thoery on symmetry , what the plane can be described as as a function of N(n).

### Re: After Reimann

All we are doing is making the primes predictable and finding how this induces the natural numbers to behave. we do have some theory's that correlate primes with naturals so inverting them would be easy. Once we have enough of these theorems we see how the inverted naturals are behaving each time and see if there is a predictable pattern.

### Re: After Reimann

Are you still there wtf?

On the convergence;

This is not the seive of erastothese.

I remove numbers with two prime factors first like 15 = 3 X 5, not numbers with two as a factor like 6 = 2 X 3

On the convergence;

This is not the seive of erastothese.

I remove numbers with two prime factors first like 15 = 3 X 5, not numbers with two as a factor like 6 = 2 X 3

### Re: After Reimann

Good morning. In my timezone (west coast USA) I'm just getting started for the day.Moyo wrote:Are you still there wtf?

On the convergence;

This is not the seive of erastothese.

I remove numbers with two prime factors first like 15 = 3 X 5, not numbers with two as a factor like 6 = 2 X 3

I see what you're doing, but isn't it a stretch to say that every possible theorem about primes has some kind of geometric form in the infinite-dimension "factor space?" It's not clear to me that there's any kind of clue here about the distribution of primes.

Basically you have a plane containing all the 2-composites (numbers with 2 prime factors); and a cube with the 3-composites; and a 4-cube with the 4-composites, and so forth. I don't see any geometric forms associated with theorems.

Then again, Wiles used advanced algebraic geometry to solve Fermat's Last Theorem. So I'm not saying there's no merit in your idea at some level; just that I don't see it.

You are correct that the factorization properties of integers are related to subtle geometric relations; but it's a long way from that general idea to anything specific.

### Re: After Reimann

Take the prime counting dunction Pi(x).

It tells us

1. What the number of primes less than a given value x.

2.What is the value of the number of naturals below a given prime.

Would be its inverse keeping track of the signs (+ or -)

Since in the infinityDimensional plane i created

The value for 1 would be x. I.e. the xth prime number is the prime we are reddering to.

What would 2 be.

We would need a new equation for two and we see how it is drawn as a line

It tells us

1. What the number of primes less than a given value x.

2.What is the value of the number of naturals below a given prime.

Would be its inverse keeping track of the signs (+ or -)

Since in the infinityDimensional plane i created

The value for 1 would be x. I.e. the xth prime number is the prime we are reddering to.

What would 2 be.

We would need a new equation for two and we see how it is drawn as a line

### Re: After Reimann

Yes, that's true. Pi(5) = 3, for example, since there are 3 primes less than or equal to 5, namely 2, 3, and 5.Moyo wrote:Take the prime counting dunction Pi(x).

It tells us

1. What the number of primes less than a given value x.

https://en.wikipedia.org/wiki/Prime-counting_function

I don't know what you mean by this. Pi(5) = 3 but that doesn't give any information on what the specific primes are.Moyo wrote: 2.What is the value of the number of naturals below a given prime. .

Don't know what you mean by that. First, note that the inverse of pi(x) is not a function. For example pi(5) = 3 and pi(6) = 3. So the inverse relation would input 3 and would output 5 and 6. What does this tell us? Not much. And what do you mean by +/- signs?Moyo wrote: Would be its inverse keeping track of the signs (+ or -).

The value for 1 would be x. I don't know what you mean by that. In your 2-plane you have the primes along each axis and the 2-composites such as 15, 21, 77, etc. at the lattice points. How does this refer to the x-th prime? The primes are along the axes, not in the plane.Moyo wrote: Since in the infinityDimensional plane i created

The value for 1 would be x. I.e. the xth prime number is the prime we are reddering to..

You'd have to answer that. I don't know what the question means.Moyo wrote: What would 2 be.

Give an example please in your 2-plane or your infinite-dimensional space.Moyo wrote: We would need a new equation for two and we see how it is drawn as a line

### Re: After Reimann

It means "how many natural numbers are below a specific prime.wtf wrote:Moyo wrote:

2.What is the value of the number of naturals below a given prime. .

I don't know what you mean by this. Pi(5) = 3 but that doesn't give any information on what the specific primes are.

E.g if we input the prime 5 we get the natural 4 since there are 4 natural numbers below the prime 5.

And if we input the prime 11 we get the natural number 10 since there are 10 natural numbers below the prime 11.

F(p) = p-1

Again , the output is a natural number.wtf wrote:Moyo wrote:

Would be its inverse keeping track of the signs (+ or -).

Don't know what you mean by that. First, note that the inverse of pi(x) is not a function. For example pi(5) = 3 and pi(6) = 3. So the inverse relation would input 3 and would output 5 and 6. What does this tell us? Not much. And what do you mean by +/- signs?

I reflected the prime counting function. The reflection of pi(5)=3 is not pi(3) = 5, thats why i said (+ and -). In a reflection some thisngs stay the same while the rest changes. The reflection i gave F(p) = p-1 takes into consideration all of that. While pi(n)=..., does not.

This is all cleard up by seeing that the inverse of pi(x) has naturals/ so if we input 11 in F(p) we get 10. and 10 is on the lattice.wtf wrote:Moyo wrote:

Since in the infinityDimensional plane i created

The value for 1 would be x. I.e. the xth prime number is the prime we are reddering to..

The value for 1 would be x. I don't know what you mean by that. In your 2-plane you have the primes along each axis and the 2-composites such as 15, 21, 77, etc. at the lattice points. How does this refer to the x-th prime? The primes are along the axes, not in the plane.

The value for (a better example) 11 would be a cone with center at 1 (since 1 is the intersection of the axis) and reaching 11 on both axis and an unknown limit in the lattice...probably something like7 X 7 .wtf wrote:Moyo wrote:

What would 2 be.

You'd have to answer that. I don't know what the question means.

Moyo wrote:

We would need a new equation for two and we see how it is drawn as a line

Give an example please in your 2-plane or your infinite-dimensional space.

Last edited by Moyo on Sun Oct 18, 2015 5:37 pm, edited 2 times in total.

### Re: After Reimann

Are you sure you are aware of what the prime counting function does? It only gives us the number (count) of primes and not what the specific primes are.wtf wrote:I don't know what you mean by this. Pi(5) = 3 but that doesn't give any information on what the specific primes are.

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