# Tagged Questions

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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### How to count the number of perfect square greater than $N$ and less than $N^2$ that are relatively prime to $N$?

I know a little about Euler's totient function that counts integer less than $N$ that are relatively prime to $N$. But I don't know how to modify the function for perfect square numbers, or maybe ...
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### For any $x\in \mathbb{N}$ does there exist $m\in \mathbb{N}$ such that $2x+1+2m, 2x+1+4m$ are both prime?

Could someone please give me a proof (or counter example) for this (I believe it is true): For any $x$ (Whole Number) there exists some $m$ (Also Whole) such that $2x+1+2m$ and $2x+1+4m$ are both ...
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### Why does this pattern occur when using modular arithmetic against set of prime numbers?

I have been recently playing around with number theory and going through the project Euler problems. So I am very new to a lot of these things. I apologize for not knowing how to look up my answer. ...
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### How far is the list of known primes known to be complete?

So there is always the search for the next "biggest known prime number". The last result that came out of GIMPS was $2^{74\,207\,281} - 1$, with over twenty million digits. Wikipedia also lists the ...
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### Possible divisors of $s(2s+1)$ follow up question.

This question is related to this post:Possible divisors of $s(2s+1)$. I have some follow up questions which should be a new post. I write $\psi(s) = s(2s+1)$. We showed that for every prime $s$ that ...
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### What is the equivalent statement of GRH in term of Redheffer Matrix or Farey Sequences?

We all know that Riemann Hypothesis (RH) has many equivalent statements. There is one statement which expresses RH in term of Redheffer matrix, there is another equivalent statement of RH which ...
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### Can this upper bound for $\sum_{i=1}^n \lfloor \sqrt{p_i} \rfloor, p_i \in \Bbb P$ be improved?

I would like to find the smallest possible upper bound for the following sum of prime radicals (OEIS A062048): $\sum_{i=1}^n \lfloor \sqrt{p_i} \rfloor, p_i \in \Bbb P$ This is my attempt. It ...
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### Effective estimates for k-almost primes

Given an integers $k$ and $\ell$ and a real numbers $\varepsilon>0$, define $f(k,\ell,\varepsilon)$ as the least $x_0$ such that for all $x>x_0$ the fraction of $\ell-$rough numbers up to $x$ ...
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### Prime divides $n^2 + 1 \Rightarrow$ prime doesn't divide $n$

How can I show that if a prime $p$ divides $$n^2 + 1$$ then it doesn't divide $n$?
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### Does there exist a closed-form expression for the following function?

I would like to find a closed-form expression for the function that is defined as follows: $T_{s}(x) = x^{s}(1 - x^{s}), \text{for prime } x \\ T_{s}(x) = x^{s}, \text{otherwise}$
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### Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers [closed]

As so far as usage in elementary number theory goes, what is the difference between the natural numbers, the integers, the rational numbers, the complex numbers, and the Gaussian integers?
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### Finding the product of a prime function…

If we take the primes $p_k < n$, and raise them to the highest power possible such that $(p_k)^{r_k} \le n$, what is the lower bounds on $\prod{ (p_k)^{r_k} }$? In other words, what are the ...
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### Existence of a prime number between $x$ and $y$ if $\operatorname{li}(y) - \operatorname{li}(x) = 1$

Is between $x$ and $y$ ($x < y$), there is always at least one prime number (or exactly one?) if $\operatorname{li}(y) - \operatorname{li}(x) = 1$?
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### Is this reasoning of Chinese Remainder Theorem correct?

Originally I want to prove $y^{p'} \equiv x^n + C \pmod p$ is always having integer solution for some prime $p$ and $p'$ It is given by my classmate, so I do not know if it can really be proved, but ...
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### Are there any two identical terms in this series, defined parallely to the primes? [closed]

Let $p_n$ denote $n$-th prime number and $k_n$ be sequence that is \begin{align} k_1 &= 1 \\ k_2 &= p_2 - k_1 &&( 3-1 = 2 ) \\ k_n &= p_n - k_{n-1} &&\text{( n is integer ...
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### Generate Sieve of Eratosthenes without “sieve” (generate prime set in interval)

How do I generate a list of primes based on the Sieve of Eratosthenes? I mean by excluding the divisible numbers beforehand, which is tricky for large numbers. I am an number theory amateur, but was ...
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### Turing Decryption Example

I know this exact same question exists but I am still having problems in understanding it. The following is given in the text: The message m can be any integer in the set {0,1,2,…,p−1}; in par­...
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### proving theorem about perfect powers

Im currently studying the journal entitled Perfect Powers with All Equal Digits but One theorem: For a fixed integer $l \geq 3$, there are only finitely many perfect $l$-th powers all whose digits ...
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### Enumeration of primes

Given a prime number $p$, there is an associated number $n(p)$, giving its ranking in the sense that $n(2)=1$, $n(3)=2$, $n(5)=3$ etc. Is there a closed form expression for $n(p)$ in terms of $p$?
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### The arithmetic function $\lambda(n)=(-1)^{a_1+\cdots +a_k}$

Define $\lambda(1)=1$, and if $n=p_1^{a_1}\cdots p_k^{a_k}$, define $$\lambda(n)=(-1)^{a_1+\cdots +a_k}$$ How can I see that \sum_{d\mid n}\lambda(d)=\begin{cases} 1 \,\,\text{ if $n$ is a square}\\...
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