# 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.

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### a question about the distribution of primes

Definition : we shall call $p$ a special prime $p_n$, if there is at least one prime of the form $2kp+1$, where $1 \leq k \leq n$ . it is obvious that we can call any Sophie Germain prime, a special ...
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### (soft) Can an efficient closed-form expression for $P_n$ be found? [closed]

I just read several old threads on here with people asking about formulas for primes, and what the implications of having one would be. As everyone was quick to point out, we already have a bunch, in ...
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### A question about the probability of being a prime?

If we chose a random number $a \leq N$, then, the probability for $a$ to be a prime is $\frac{1}{\log N}$. Now, if there are some primes that do not divide $a$, then what is the probability for $a$ ...
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### Why this strategy is true…

To find prime number if we want to be sure, why can't we just put odd number of zeros between two ones and get a prime number, and also why this becomes true and how to prove it. Example: 101 10001 ...
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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, $S$ ...
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### Distribution of Digits of Binary Expansion of Primes

In considering the binary expansion of prime numbers, I'm interesting in the skew of digits towards 0 or 1. I searched through other questions and arrived at: Last digits of primes I just want to ...
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### Algorithm generating subset of primes, can we classify which of them or estimate how large percent of primes are generated?

Assume I have following algorithm: Two lists of numbers, first starting at 2, second starting empty. We now follow rule: Add a number to first list which makes difference with latest number the ...
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### A ratio connected to the distribution of primes

According to the Prime Number Theorem, a number $n$, roughly speaking, has probability of primality $\sigma_n:=1/\ln n$. As every schoolchild learns, one can test the primality of $n$ by looking for ...
### Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.
Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$. I cannot give a proof to this, although I have try on ...