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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.

A prime number (or a prime) is an element of the greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number ... The fundamental theorem of arithmetic establishes the central role of primes in :

Any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

Here you get the first 50 millions of primes.

The concept of prime numbers is extended in ring theory, where an element $p$ of a ring $R$ is prime if and only if whenever $p\mid ab$, then $p\mid a$ or $p\mid b$.

One can easily see that this extends the definition of prime numbers in the natural numbers.

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Co-prime binomial coefficient

A number 'r' is prime if and only if $\binom{r-1}{k} \equiv(-1)^k \pmod r$ Since 'r' is a prime and it gives non-zero remainder by dividing $\binom{r-1}{k}$ . So $\binom{r-1}{k}$ and 'r' are co-primes If a0,a1,a2,a3,..........,ar-1 are coprimes…
hanugm
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check for primality of a number

Is 2^131 - 1 a prime number? if so how can i proof it, or if no how? In the general is there a way for primality check for a 2^n - 1 Thanks
frogatto
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About how to calculate prime numbers (additional content: add calculation methods, questions you want to ask, etc.)

Sorry for the previous question. This is my first post and I didn't write anything special because I wasn't sure if it was worth showing to others. Also, since I am not good at English, I use machine translation. Therefore, please forgive me if…
maki
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Prime numbers and their largest known difference

What is the largest difference between two consecutive prime numbers? E.g 37-31=6 So which is the largest margin that exists between two consecutive prime numbers?
Nelson Njagi
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What's known about primes of the form $m^n+1$?

For example, the Fermat primes are primes of the form $2^{2^n}+1$. I'm wondering if the primes $m^n+1$ have a name. More importantly, I'm wondering if there are tables of these primes, and what else is known about them. I'm hoping that someone can…
Matt Groff
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Perfect square, and prime number and positive integer

Find all pairs of numbers $(p,n)$ in which $p$ is a prime number, $n$ is a positive integer such that $p^n + 144$ is a perfect square.
Anh Phùng
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Legendre's constant

There is a somewhat funny article on Wikipedia called Legendre's constant. In it is stated that there supposedly would exist a unique value $B = 1$ such that $$ π(x) \sim \frac{x}{\ln(x) - B}. $$ But first of all, the prime number theorem states…
Cloudscape
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Big nth prime numbers

I know the following nth prime numbers: $n = 10 \longrightarrow n$th prime $= 29$ $n = 10 ^ 2 \longrightarrow n$th prime $= 541$ $n = 10 ^ 3 \longrightarrow n$th prime $= 7,919$ $n = 10 ^ 4 \longrightarrow n$th prime $= 104,729$ $n = 10 ^ 5…
Clyde Tillman
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Upper bound for the largest number of consecutive integers such that each is divisible by primes less than the nth prime

Let a(n) be the largest number of consecutive integers such that each is divisible by a prime <= the n-th prime. On oeis.org : https://oeis.org/A058989 there is an upper bound mentioned which is $ a(n) < $${n^2} \cdot {(ln(n))^2}$ This upper bound…
Kim Hollesen
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If $K$ and $S$ are prime numbers then how can I prove that for some $n$, there exist prime numbers of the form $K+2n+2$ and $S-2n$?

If $K$ and $S$ are prime numbers then how can I prove that for some $n$, there exist prime numbers of the form $K+2n+2$ and $S-2n$?
skrstars
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Are there alternatives for the prime indicator function $ \mathbb{1}_{n \in \mathbb{P}}=-\mu(n) \frac{\Lambda(n)}{\log n} /(\cos \pi n)^2 $?

I have found in the preprint An Exact Formula for the Prime Counting Function the following expression for a function that is $1$ whenever n is prime, and $0$ otherwise: $$ \mathbb{1}_{n \in \mathbb{P}}=-\mu(n) \frac{\Lambda(n)}{\log n} /(\cos \pi…
R. S.
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How can I prove that $\frac{x}{\ln(x)} < \pi(x)$ for $x \geq 17$?

I saw on Wikipedia that $\frac{x}{\ln(x)} < \pi(x)$ for $x \geq 17$ but it doesn't provide the proof. What is the proof for this and are there any legitimate papers that I can cite?
rokkc
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Is this function good at finding the nth prime? What is it's big O?

$$f\left(m\right)=\sum_{i=0}^{m^{2}}⌊cos\left(⌊ \frac{\sum_{o=1}^{i}\left(\prod_{n=2}^{⌊\sqrt{o}⌋}\left(1-⌊\cos\left(\pi\frac{o}{n}\right)^{2}⌋\right)\right)}{m+1}⌋\right)^{2}⌋$$ I was watching a video about how a formula for finding the nth prime,…
Benjamin Trumpy
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Is it prime or composite

Is $1359^{73}+(137^{18}+253^{16}+154^{22})^{19}$ prime or composite? and WHY? For example for $17^{13}+34^{19}+51^{23}$ i do this: $17(17^{12}+2×34^{18}+3×51^{22})$ Then obviously it is a composite number(can devided by 17). or in some question if…
Alireza
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Prime factorization rules

When you have a number like 81. Is it safe to assume that if the number can't be divided by 2 or 3 that it's prime if it ends with a 1?
Liger86
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