# Some Flat Equation Prime Number Aproximations

• qpwimblik
In summary, the author is investigating approximating Pi(n) and estimates Pn to be between 0.075 and 0.5 n of Pi.
qpwimblik
for estimating Pn

1/2*(8-8.7*n-n^2+1/2*(2*abs(ln(n)/ln(3)+ln(ln(n)/ln(2))/ln(2))+abs((ln(ln(3))-ln(ln(n))+2*n*ln(ln(n)/ln(2))+sqrt(((8*ln(3)*ln(n))/ln(2)-ln(ln(2))+ln(ln(n)))*ln(ln(n)/ln(2))))/ln(ln(n)/ln(2))))*(-1+abs(ln(n)/ln(3)+ln(ln(n)/ln(2))/ln(2))+abs(-(1/2)+n+sqrt(((8*ln(3)*ln(n))/ln(2)-ln(ln(2))+ln(ln(n)))*ln(ln(n)/ln(2)))/(2*ln(ln(n)/ln(2))))))

or

1/2*(3-(8+ln(2.3))*n-n^2+1/2*(-1+abs(-(1/2)+n+sqrt(ln(ln(n)/ln(2))*(-ln(ln(2))+ln(ln(n))+(8*ln(3)*ln((n*ln(8*n))/ln(n)))/ln(2)))/(2*ln(ln((n*ln(8*n))/ln(n))/ln(2))))+abs(ln(n)/ln(3)+ln(ln((n*ln(8*n))/ln(n))/ln(2))/ln(2)))*(2*abs(ln((n*ln(8*n))/ln(n))/ln(3)+ln(ln((n*ln(8*n))/ln(n))/ln(2))/ln(2))+abs(1/ln(ln(n)/ln(2))*(ln(ln(3))-ln(ln(n))+2*n*ln(ln(n)/ln(2))+sqrt(((8*ln(3)*ln(n))/ln(2)-ln(ln(2))+ln(ln((n*ln(8*n))/ln(n))))*ln(ln((n*ln(8*n))/ln(n))/ln(2)))))))

And for approximating the Pi(n)

1/(3*abs(ln(n)))*((2-n*(n-ln(n))+(-1+abs(n)+abs(ln(ln(n)))/ln(pi))*(abs(n)+abs(ln(ln(n)))/ln(pi))-(2*ln(ln(n)))/ln(pi))/(1+abs(ln(n)))+1/(abs(ln(n))+ln(2))*ln(2)^2*(2-n*(n-ln(n)/ln(2))+(-1+abs(n)+abs(ln(ln(n)))/ln(pi))*(abs(n)+abs(ln(ln(n)))/ln(pi))-(2*ln(ln(n)))/ln(pi))+1/(abs(ln(n))+ln(3))*ln(3)^2*(2-n*(n-ln(n)/ln(3))+(-1+abs(n)+abs(ln(ln(n)))/ln(pi))*(abs(n)+abs(ln(ln(n)))/ln(pi))-(2*ln(ln(n)))/ln(pi)))

I do apologise If you want more advanced formula's with Summation's and or Factorial's and or big O notation but I don't understand that stuff very well.

we are more interested to know how you derived those equations.

Well I was investigating counting with A002260 and A004736 and tried using The triangle number formula to balance both pattern one and pattern two, the formula to which is on my website, Then I Played with the counting function with some other functions like log and then simplified with the help of the computer my best effort at predicting Pn and tweaked around with that until well what you see so far as my effort.

I still have my work cut out for me to see how accurate I can get and am working towards a few tweaks with the addition of some extra functionality including making the smooth curve reverberate. All still as a flat equation.

The pi(x) approximation is my first attempt (roughly 2 hours work) at that problem so it should be quite easy for me to Improve it's accuracy and give it the Pn weaving ability that the far more worked on Pn estimates have.

Last edited:
My Pn estimates seem weave round Pn with an upper bound of 0.075 n of Pn and a lower bound of 0.5 n of Pn all at an ever slower speed exponentially relative to Pn, which Means with a slight modification to the formula I can get with in roughly 0.28 n of Pn for any number no matter how large providing this conjecture works to infinity.

I would suggest you post some numerical results and/or graphs to let people get a feel for how good your approximations are.

on it way I should have that for you soon.

## What is a "Some Flat Equation Prime Number Aproximation"?

A "Some Flat Equation Prime Number Aproximation" is a mathematical concept that involves using a simple equation to approximate or estimate prime numbers. It is a way to quickly identify potential prime numbers without having to manually check each number for factors.

## Why use a flat equation for prime number approximations?

A flat equation is used for prime number approximations because it is a simple and efficient way to identify potential prime numbers. It involves using a basic mathematical formula, such as n^2 + n + 41, to generate a list of numbers that may be prime. This saves time and effort compared to manually checking each number for factors.

## How accurate are "Some Flat Equation Prime Number Aproximations"?

The accuracy of "Some Flat Equation Prime Number Aproximations" depends on the specific equation used. Some equations, such as n^2 + n + 41, are highly accurate and can identify a large number of prime numbers. However, others may have limitations and may not be as accurate. It is important to carefully choose the equation based on the intended purpose.

## What are the potential applications of "Some Flat Equation Prime Number Aproximations"?

"Some Flat Equation Prime Number Aproximations" have several potential applications, including cryptography, data encryption, and data compression. They can also be used in computer algorithms and programs that require prime numbers for various calculations.

## What are the limitations of "Some Flat Equation Prime Number Aproximations"?

While "Some Flat Equation Prime Number Aproximations" are a useful tool, they have some limitations. They may not be able to identify all prime numbers, and some equations may only work for a limited range of numbers. Additionally, they may not be as accurate as other methods, such as using the Sieve of Eratosthenes, for identifying prime numbers.

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