Linked Questions

71 votes
4 answers
59k views

Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
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14 votes
2 answers
3k views

Who discovered the first explicit formula for the n-th prime?

I just found out on Wolfram that there is a formula for the n-th prime in terms of elementary functions. I wonder who found it and if he was rewarded for this. The formula (here) is: Also shown at ...
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  • 1,367
9 votes
3 answers
4k views

What would be the immediate implications of a formula for prime numbers?

What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
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2 votes
1 answer
2k views

Formula for the nth prime number. [closed]

I need to see how to make a certain development, and did not find any reference to do, the only quote I found, so is the following, I know (have shown) $$ \pi(m)=-1+\sum_{j=1}^{m} F(j) $$ with $$F(j)=...
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  • 3,623
3 votes
1 answer
625 views

A Shorter Proof of Rosser's Theorem Without Using The Prime Number Theorem

While researching on the elementary proof of Bertrand's Postulate I came to know about a theorem of Rosser's which states that $p_n$ $>$ $n$ $\text{ln}$ $n$. I have seen Rosser's original proof and ...
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7 votes
2 answers
449 views

a practical prime counting function

Looking at Prime counting functions on Wikipedia, I only found formulas with no hint on how people got there. So, to better understand, I've decided to build one from scratch, starting from a naive ...
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3 votes
1 answer
726 views

A formula that counts exactly the twin prime averages occuring in an interval $[a,b]$ is surprisingly succinct!

Let $p_n$ denote the $n$th prime number. Let $p_n \lt a \lt b \lt p_{n+1}^2$ be any such integers. Their oddness or divisibility does not matter as in my previous posts, which makes this formula ...
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10 votes
0 answers
543 views

Why does this identity equal the number of primes?

Can someone explain why this identity gives the number of primes? I don't understand it. $D_{0,a}(n) = 1$ $D_{k,a}(n) = \displaystyle\sum_{j=1}^{k} \binom{k}{j}\sum_{m=a+1}^{\lfloor n^{\frac{1}{k}}\...
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1 vote
2 answers
158 views

What's a "formula"? Is there a "formula" for the n'th prime number?

Talking to some undergraduates at my university, the idea came up that there was a "formula" for the n'th square number (the formula is $n^2$) but there was no "formula" for the n'...
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1 vote
1 answer
294 views

How to calculate the $i$-th element in the sequence of prime numbers?

The sequence of prime numbers is the set of prime numbers in their natural order (that is, $2, 3, 5, 7, 11, 13, 17,...$). The German wikipedia entry on sequences states the following: Given $i$, ...
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  • 323
2 votes
2 answers
155 views

Idea for primality testing based on a trigonometric product

This is an idea that I had about 3 months ago. I tried some college professors, they didn't care. I tried to solve, but with no luck. I ask for help to find the closed form of the following product ...
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3 votes
1 answer
124 views

Is $\lim\limits_{m\to\infty}\frac{1}{m}\sum_{n=1}^m \cos\left(\frac{2\pi nj}{s}\right)$ "useful"?

In this answer it is explained that a big reason that the nth prime formula discussed with (13) and (14) isn't "useful" is because $\lfloor\frac{x}{b}\rfloor-\lfloor\frac{x-1}{b}\rfloor$ ...
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  • 3,193
0 votes
0 answers
112 views

Inquiry on prime counting function

One of my close friends and I have been working towards an exact prime counting function. The approach we have came up accurately produces the number of composite numbers that occur before a given ...
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  • 63
2 votes
0 answers
71 views

Is there any proof in Number theory claimed or dis-claimed the existence of universel form generating primes?

Many students , teachers of mathematics provide a huge efforts to get a universal closed form of a function which generate prime number , Really i'm very interesting to know if there is any proof ...
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