# 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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### Conjecture on Infinitely Many Consecutive Pairs of Early Primes

An early prime is one which is less than the arithmetic mean of the prime before and the prime after. Conjecture: There are infinitely many consecutive pairs of early primes MY attempt Well, the fact ...
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### The prime race $a_i \mod 11$ vs $b_i \mod 11$ conjecture

Let $f(n,a)$ be the number of primes of type residue $a \mod 11$ between $1$ and $n$. Is it true that for all $n>1$ we have f(n,1)+f(n,2)+f(n,3)+f(n,5)+f(n,7)+f(n,6)+f(n,8) > f(n,4)+f(n,9)+f(n,...
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### Collatz conjecture and prime numbers [closed]

With the intention of understanding how prime numbers contribute to the numerical results we get when we perform any possible numerical calculation (also on real numbers), since they are those natural ...
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### About the solutions of $\dfrac{x^p - y^p}{x - y} = a^2+pb^2$

Using theorem $IV$ from this article, is possible to prove that when $p$ is a prime $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the equation $\dfrac{x^p - y^p}{x - y} = a^2+pb^2$ ever ...
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### Is this conjecture about twin primes known to be false?

I'm not sure if this has been investigated before. This is a kind of strong twin prime conjecture Define a first twin prime as the lower of a twin prime pair, while a second twin prime is the upper of ...
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### Counting odd integers in a consecutive sequence divisible by a given prime

For any integer $a, n > 1$, let $O_p(a,a+n)$ be the count of $i$ such that $a < i < a+n$, $i$ is odd, and $p | i$ where $p$ is any odd prime. Does it follow that for any such $p, a, n >1$, ...
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1 vote
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### Every even number is the sum of at most three primes

I'm failing to find online references to the following problem, which to me seems a slight weakening of the Goldbach conjecture. Conjecture: every even integer $n$ is the sum of at most three primes. ...
1 vote
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### Expected number of factors of $LCM(1,…,n)$ (particularly, potentially, when $n=8t$)

I’m trying to prove something regarding $8t$-powersmooth numbers (a $k$-powersmooth number $n$ is one for which all prime powers $p^m$ such that $p^m|n$ are such that $p^m\le k$). Essentially, I have ...
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### The sum $f(x)= \sum_{p \le x} e^{\frac{1}{p\log p}}$ is very close to $\pi(x)$. Why is that?

The prime counting function $\pi(x)$ counts the number of primes less than a given $x$. There are other counting functions like Chebyshev's functions which count sums of logarithms of primes up to a ...
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1 vote
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### What's The Minimum Number Of Prime Factors Needed To Replace "3x+1" With Any Linear ("mx+b") Function And Still Work Like The Collatz Conjecture?

Apologies; I know there are a few assumptions used to pose this question, namely: 1): That yes, any mx+b function can work like the infamous "3x+1," problem... ...Provided, that you give it ...
1 vote
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### Generating function of partitions of $n$ in $k$ prime parts.

I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$. I know ...
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### Proving there are infinitely many primes using factors of Fermat numbers [duplicate]

Is this proof acceptable? Theorem (Lucas) Every prime factor of Fermat number $F _ n = 2 ^ {2 ^ n} + 1$; $(n > 1)$ is of the form $k2 ^{n + 2} + 1$. Theorem The set of prime numbers is infinite. ...
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### For each integer $k,$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},$ s.t. for each $n,\ p_{n+1}=2p_n- 1$ or $p_{n+1} =2p_n+1?$

For each $k\in\mathbb{N},$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},\$ such that for each $n,$ the following is satisfied: $p_{n+1} = 2p_n- 1\$ or $p_{n+1} = 2p_n + 1?$ If yes then ...
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1 vote
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### Primes of the form $3^a+2^b$ or $3^a-2^b$

I was searching for information about prime numbers, and somehow, I ended up verifying whether certain numbers are prime. I seem to have stumbled upon some far-fetched relationships with the following ...
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