# 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 Prime Numbers and Exponential Equipment

How to generate a rigorous proof for - The specific conjecture which states that we can always frame every Prime number in an exponentially framed equation with perfect cubes & squares. I'll ...
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### Prove $r$ the smallest quadratic non-residue modulo $p \geq 3$ is prime

I've been struggling to find the solution for this question for a while now and thought I might as well ask for some help. The question is: Let $r$ be the smallest positive quadratic non-residue ...
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### $π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

Thinking about the prime number theorem, I wondered whether it is known that there is some constant $c$ such that $π(x+y) ≤ π(x) + c·y/\ln(y)$ for every integers $x,y > 1$. I read that experts ...
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### Ordering sequences of prime decompositions

C.f. Engel's Problem Solving Strategies; p. 132, #33: Out of $n+1$ integers $\leq 2n$, find $p,q$ s.t. $p|q$. Rather than soliciting a solution I'm wondering if there's a way I can make my attempt ...
1answer
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### Error term of the prime number theorem on GRH?

I know that under GRH we have $$\sum_{\substack{1 \leq n \leq X \\ n \equiv a (q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} (\log X^2)).$$ From this I would like to deuce a abound for $E$ ...
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### Are there forced factors of numbers of this kind?

Let $$f(n):=n^{n^2}+(n+1)^{(n+1)^2}$$ for a positive integer $n$. For clarification $n^{n^2}$ means $n^{(n^2)}$ , analogue for the other summand. Can we find a concrete factor of $\ f(n)\$ (like ...
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### Complex numbers $z$ such that $n-z$ and $n+z$ are Gaussian primes

Is it known whether for all large enough positive integer $n$, there exists a complex number $z$ such that $\vert z\vert<n$ and both $n-z$ and $n+z$ are Gaussian primes? If yes, are there results ...
1answer
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### Is it true that for all large enough integer $n$, there is a integer $m$ such that $n+im$ is a Gaussian prime?

Let $n$ denote a positive integer greater than $6$. Is it known whether there is always an integer $m$ such that $n+im$ is a Gaussian prime?
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### Is this sum convergent or divergent? [duplicate]

Is the sum $$\sum_{p\text{ prime}} \frac{1}{p\ln(p)}$$ convergent or divergent ? It is well known that the sum $$\sum_{p\text{ prime}} \frac{1}{p}$$ is divergent but very slow divergent. The above ...
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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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### Use a pairing function for prime factorization

Let $n$ be the product of the two unknown primes $a$ and $b$ ($a < b$): $ab = n$ My idea was to use a pairing function so: $a = f(p)$ and $b = g(p)$ Then the equation is: $f(p) \cdot g(p) = n$ ...
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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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