# Questions tagged [prime-numbers]

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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### Every sufficiently large positive integer is the average of $n$ distinct primes for certain $n \geq 2$?

I want to generalize a stronger Goldbach's conjecture a little bit because that might help solve it. I was thinking: For all $n \geq 2$, every sufficiently large positive integer $x \geq b_n$ is ...
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### Checking for prime constellation

How does one systematically check if a given configuration of prime numbers $p_1, p_2, ... p_n$ is the densest possible configuration of primes in the range $[p_1, p_n]$? (The densest configurations ...
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### Get numbers that have only 2,3 and 5 as prime factors

I am given an integer N. I have to find first N elements that are divisible by 2,3 or 5, but not by any other prime number. ...
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### Found a new way to do CRT on prime vector

I found a new way to do CRT on prime vector. Given prime list P, and some residuals R=mod(n,P) , for example: P=[2 3 5 7 11] R=[1 0 3 1 9] This matlab function ...
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### Are there an infinite number of primes which are any multiple of $n$ apart? [closed]

Are there an infinite number of primes which are any multiple of $n$ apart? That is take $n\in \mathbb{N}$, then is there an infinite number of primes which are separated by $\textbf{any}$ of the ...
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### Prime counting function formulas

Are there any elementary (including floor, ceiling, mod) representations of the prime counting function. Or one without an integral.
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### Is the additive rational group $\mathbb{Q},+$ generated by $\frac{1}{p}$ where p is a prime?

So it is known that the additive group of rationals numbers $\mathbb{Q},+$ is generated by $\frac{1}{n}$ with $n \in \mathbb{N_0}$ so that: $$\mathbb{Q},+ =grp\{\frac{1}{n} | n \in \mathbb{N}\}$$ Now ...
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### 4 distinct integers with prime sum for each triple

Here is a nice high school olympiad math problem: Can you choose 4 distinct positive integers so that the sum of each 3 of them is prime? How about 5? It looks that just by looking at reminders mod ...
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### Solutions of $x^n=a$ modulo primes.

Let $p$ be an odd prime and $a$ be an integer that is not a perfect square. We can impose a congruence condition on $p$ that guarantees the equation $x^2=a \pmod p$ does not have a solution. For ...
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### First few smallest Carmichael numbers congruent to $11 \pmod {12}$

There are known to be infinite Carmichael numbers congruent to $a\pmod b$ for coprime integers $a$ and $b$. There are plenty of examples of small Carmichael numbers congruent to $1, 5, 7 \pmod {12}$, ...
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### prime divisor of $3n+2$ proof

I have to prove that any number of the form $3n+2$ has a prime factor of the form $3m+2$. Ive started the proof I tried saying by the division algorithm the prime factor is either the form 3m,3m+1,3m+...
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### CRT on prime vector

I have a question that intrigue me: Given primes and some reminder vector P=primes(13)=[2 3 5 7 11 13] R=[1 2 1 3 9 11] (R=mod(1571,P)) What option do i have ...
534 views

### How to make a pair of six-sided dice whose sum is always a prime number?

In other words, how can I find two sets of six distinct integers $a_1, \dots, a_6 \in \Bbb Z$ and $b_1, \dots, b_6 \in \Bbb Z$ such that $a_i+b_j$ is prime for any $i, j \in \{1, \dots, 6\}$?
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### Proof that $\sqrt{17}$ is irrational

Consider $\sqrt{17}$. Like the famous proof that $\sqrt2$ is irrational, I also wish to prove that this number is irrational. Suppose it is rational, then we can write: $$17 = \frac{p^3}{q^3}.$$ ...
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### How many divisors of $\phi(m)$ do not divide $m-1$?

Lehmer's totient problem asks if there exists a composite number $m$ such that $\phi(m)$ divides $m-1$. Lower bounds on $m$ has been established but we do not know if a solution exists. Clearly, if we ...
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### How dense are primes congruent to 1 and 3 (mod 4)? [duplicate]

There are infinitely many primes of the form $4n+1$ and $4n+3$. In a given interval $[0,N]$ for a large enough $N$ do we expect to see the same number of primes congruent to $1$ and $3$ (mod 4)?
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### Is there an Efficient Way to Divide by a Mersenne Prime?

Mersenne primes are used in Computer Science and Cryptography because they support fast modulo computation. If $p$ is a Mersenne prime, $n \bmod p$ can be computed with just a few add and shift ...
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### Wolstenholme Number

A Wolstenholme number is the (reduced) numerator of the fraction $1+{1\over4}+\cdots+{1\over n^2}$. The first few are $1, 5, 49, 205, 5269, 5369, 266681, 1077749$. Are there Wolstenholme numbers that ...
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### Proof of the theorem

I have been unsuccessful in trying to find a proof of this theorem the name of which, if existing, I don't know. For any prime number $p$ and natural number $A$, where $p$ doesn't divide $A$, the ...
The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer pairs whose variances are equal; the smallest such pair is $(691, 817)$. However I ...