# 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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### Characterization of primes $(6n+1, 6n-1)$ that are not twins

According to OEIS Sequence A002822(https://oeis.org/A002822), it states that $6n+1$ is a twin prime $iff$ $n$ is not of the form $6ab \pm a \pm b$. I was wondering if anyone had a proof for this. ...
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### Proof of primes of the form $6n+1$

According to OEIS Sequence A002476 (https://oeis.org/A002476), it says that all primes of the form $6n+1$ can be written in the form: $x^2 - xy + 7y^2$ with $x$ and $y$ non-negative. I was wondering ...
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### Finding $1/x^2 + 1/x^3 + 1/x^5 + \dots$

The following function came up in my work: $$f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots.$$ Naturally, this converges for ...
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### Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
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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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### Primes is in P, proof of hendrik Lenstra Jr. lemma

In the paper describing AKS primality test : http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n2-p12.pdf On page no. 8 Lemma 4.7 last paragraph, I cannot understand how number of ...
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### Is the sum of coefficients 2?

Is the sum of the coefficients of the polynomial interpolation of the data $(1,p_1),(2,p_2),...,(n,p_n)$ for some positive integer $n$ (where $p_n$ is the $n$th prime) always equal to two? I've ...
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### A question about arithmetic progressions and prime numbers

I took number $3$ and observed: $3$ is an arithmetic progression of length one. $3,5$ is an arithmetic progression of length two. $3,5,7$ is an arithmetic progression of length three. Then I took ...
If $n$ is prime, then $n$ is not divisible by any prime number between 1 and $\sqrt{n}$ strictly. (Assume that $n$ is a fixed integer that is greater than 1.). I searched online and found that "...