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Questions tagged [distribution-of-primes]

Use this tag for questions related to the branch of number theory studying distribution laws of prime numbers among natural numbers.

The central problems are to find the best expression as $x \to \infty$ for–

  • the number of prime numbers not exceeding $x$, and
  • the number of prime numbers not exceeding $x$ in an arithmetic progression.
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Are there primes of every possible number of digits?

That is, is it the case that for every natural number $n$, there is a prime number of $n$ digits? Or, is there some $n$ such that no primes of $n$-digits exist? I am wondering this because of this Project Euler problem:…
ET-phone-homology
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Prime numbers the rank of which is also a prime.

$127$ has an interesting property: It is the $31$st prime number and its rank ($31$) is also a prime. $31$ is the $11$th prime so its rank is also a prime. $11$ is also a prime number with a rank ($5$) that is also a prime. $5$ is the 3rd prime…
Pan Pap
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The famous prime race and generalizations

So I was messing around with the famous prime race that comes down to this: We make a list of primes. The list has two rows; the top row is for primes $1\mod 4$ and the bottom row for primes $3\mod 4$. Our list, up to the 10th prime: 5 13 17 …
user304329
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Why do number theorists care so much about how well $\text{Li}(x)$ approximates $\pi(x)$ if it's not our best approximation?

An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(x)$ that converges, upon addition of the…
Tanishq Kumar
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What is the link between Primes and zeroes of Riemann zeta function?

Usually, the Riemann hypothesis is introduced along the following lines: (1.1) Geometric progressions were known for forever (1.2) Euler factorization links a product of primes to a sum of natural numbers (1.3) The harmonic series diverges, thus…
sixtytrees
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Stronger result than bertrand's postulate

It is well known that there is a prime number between $n$ and $2n$ for all $n$. I decided to go deeper: is there a lower bound on the number of primes between $n$ and $2n$ for "large enough" $n$? For instance, I found empirically that…
Joseph Martin
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A rash guess about distribution of primes based on meager empirical evidence?

Between the prime numbers $n=1327$ and $n+k = 1327+34 = 1361$ there are $k-1=33$ consecutive composite numbers. If you double those bounding primes, getting $2\times1327=2654$ and $2\times1361=2722$, then between them you find $14$ primes, i.e. in…
Michael Hardy
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Are there infinitely many primes less than $q^{1+\epsilon}$ equivalent to $1$ mod $q$?

Fix $\epsilon>0$. As $q$ becomes large, is it true that the number of primes less than $q^{1+\epsilon}$ congruent to $1$ modulo $q$ will tend to infinity? A conjecture of Montgomery says that the number of primes congruent to $a$ mod $q$ should tend…
Milo Moses
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A prime generating algorithm

I was trying to explain the famous proof of infinitude of primes to a young one, and I tried to explicitly show some examples. So, I said something like Let the only primes be $2,3,5$. Then $$N=2\times 3\times 5+1=31$$ which is a prime. So, let the…
Sayan Dutta
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Does the truth of one imply the other? A simple Collatz generalization in terms of primes.

Let $f_i:\mathbb{N} \to\mathbb{N}$. The Collatz function states that the following iterated map will eventually equal to 1: $$f_0(n) = \begin{cases} n/2, & \text{if}\ 2\mid n\\ 3n+1, & \text{otherwise} \\ \end{cases}$$ Noting that $2$ and $3$ are…
Math777
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Density of Numbers with Exactly One Prime Factor of Multiplicity 1

Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ does not contain the prime numbers). What is the…
Ashvin Swaminathan
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Prime Counting Function from the Sieve of Eratosthenes

It is known that the Sieve of Erastothenes can be analytically stated as: Let $P$ be the product of the prime numbers $\leq \sqrt{N}$ and $\omega(n)$ the number of different prime divisors of $n\in\mathbb N$. Then $$ \pi(N)=\pi(\sqrt{N})-1+\sum_{d…
user3141592
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Possible Riemann's Hypothesis proof?

First of all, I imagine it will not be correct, just because of its simplicity, but I would also want to know why, as I can't find any mistake on it. The "proof" would be based on convining two main theorems/formulae. The first one, would be this…
user3141592
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Is this a regularity in primes?

For any prime $p$ subtract $24$ continuously. The last value before $0$ will always be one of these $8$ primes: $\{ 1, 5, 7, 11, 13, 17, 19, 23 \}$. Prime Distribution Across Lengths of 24 Primes in Blue. Root Prime Path in Red As can be observed…
Iapyx
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Collinear primes $(n,p_n)$

This is about prime numbers such that at least three points $(i,p_i)$, $(j,p_j)$ and $(k,p_k)$ are on the same straight line. Conjectures: For any pair $(i,p_i),\, i>1$, there are two different primes $p_j,p_k>p_i$ such that $(i,p_i)$, $(j,p_j)$…
Lehs
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