# 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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### The eventual advantage of a primality test without known exceptions

The primality test of Fermat with base $2$ seems to be as secure as the computer hardware for testing numbers big enough. However, I think there are an infinite numbers of false primes using this ...
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### Is the Fermat primality test secure enough for very big numbers?

The random variable $X_m$ is the number of trials before $n\notin\mathbb P\wedge n|2^{n-1}-1$ where $n$ is an odd random integer $2^{m-1} < n < 2^m$. Computer simulations makes me believe ...
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### Proving a number is Carmichael

here is my question: Let $p>3$ be prime, s.t $q = 2p-1$ and $g = 3p-2$ are primes as well. (For example $p=19$,$13$,$7$). Prove that $N = pqg$ satisfies $p-1|N-1$, $q-1|N-1$ and $g-1|N-1$. I ...
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### Describe the prime elements of the ring $\mathbb Z[\sqrt{-2}]$ [duplicate]

I have a ring $\mathbb Z[\sqrt{-2}]$ and I need to describe all the prime numbers of that ring. How I can do that? Thank you
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### What are the different ways to get a first-order formula that express the statement“$P$ is the $n$-th prime”

I know that such a $2$-predicate formula exists since Enderton's have already constructed such a formula in his text on mathematical logic but it was not easy to remember so I wonder if there is other ...
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### What is the first 21-digit prime number after the decimal point of Pi

We know the recent computers are 64-bit, and the maximum integer number is 18446744073709551615, whether you can find the first 21-digit prime number after the decimal point of $\pi$? Please show me.
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### What is meant by strictly in this statement?

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 "...
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### A question about distribution of prime numbers

For prime numbers $p$, let $f(p) = \min\{k: p+k \text{ is prime}\}$. (BTW, is there a standard notation for this function?) For $p=113$ we have $f(p) = f(113) = 14$, i.e. the next prime number above ...
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### A rash guess about distribution of primes based on meager empirical evidence?

Between the prime numbers $n=1327$ and $n+k = 1327+34 = 1361$ there are $k-1=33$ consecutive composite numbers. If you double those bounding primes, getting $2\times1327=2654$ and $2\times1361=2722$, ...
### Probability distribution of $\omega'(n)$. [duplicate]
$\omega(n)$ is the number of distinct prime factors of $n$ and $\omega'(n)$ is the number of distinct prime factors of $n$ with multiplicity. For example if $p,q$ are prime numbers then \$\omega(p^2q)=...