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:…
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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…
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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 …
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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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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…
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Prime Counting Function from the Sieve of Eratosthenes

This may seem a very simple question, but I did not find any answer for it on the Internet. It is known that the Sieve of Erastothenes can be analytically stated as: $$\pi(x)-\pi(x^{\frac{1}{2}})+1=\sum_{d} (-1)^{\nu(d)} \left \lfloor \frac {x}{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…
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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…
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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…
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What is the link between Primes and zeroes of Riemann zeta function?

Usually Riemann hypothesis is introduced along this lines. (1.1) Geometric progressions were known since forever (1.2) Euler factorization links a product of primes and a sum of natural numbers (1.3) Harmonic series diverges, thus there are infinity…
sixtytrees
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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…
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Do lucky numbers contain arbitrarily long arithmetic progressions?

The lucky numbers are defined by a sieve, which results in numbers that asymptotically mirror the prime density $\sim n / \log n$: $$ 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, \ldots $$ OEIS…
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Is the product of primes less than $3\log_2{n}$ always at least $n$?

Consider the product of all primes less than $3 \log_2{n}$. Is it true that this product is always at least $n$ for all positive integers $n$? In general, what is the smallest $x_n$ so that the product of all primes less than $x_n$ is always at…
graffe
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Shows that there are 1000 consecutive positive integers containing exactly 10 prime numbers

As the title says, how can i demonstrate that? I know how to make a sequence of n consecutive positive integers that are composed. Let be k the length of the sequence, and consider the following consecutive numbers: $(k+1)!+2,\ (k+1)!+3,\ ...\ ,\…
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Subsets of primes with unevenly distributed remainders mod 4

I am an undergrad math major and have been doing some research as a hobby. In short, I am looking at primes whose squares divide some number of the form $n^n + (-1)^n (n-1)^{(n-1)}$. I have found that for primes up to 800,000, about 55% of them are…
Will Craig
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