# 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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### Remainder of Fermat's little theorem sum

With p being a prime number, what is the remainder of $$\sum_{k=1}^{p-1} {k^{p-1}}$$ divided by p ? I know that Fermat's little theorem states that for a prime number p, and a number A that is not ...
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### Can we remove any prime number with this strange process?

This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post. The process goes as follows: Start with the ...
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### Maximum $k$ for odd consecutive primes $p,q$ and $q-p=2k$ for $1$ to $k$ [on hold]

For consecutive odd primes $p, q$ with $q-p=2\cdot k$ for consecutive numbers $1$ to $k$, is some maximum $k$ known? Is there a table showing at what least prime each gap occurs? Thus one would want ...
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### If there exist infinitely many $x \in \mathbb{Z}:3x^2+3x+1 = 3p-2$ for $p \in \mathbb{P}$, show there exist infinitely many $y:3y^2+3y+1$ is prime

Assume there exist infinitely many $x$ such that: $$3x^2+3x+1 = 3p-2$$ Where $p$ is prime. Can it be shown there exist infinitely many $y$ such that: $$3y^2+3y+1=q$$ Where $q$ is prime? I believe ...
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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 ...
2answers
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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 ...
5answers
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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 ...
1answer
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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.
2answers
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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 ...
1answer
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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 ...