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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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Is it possible to predict the number of primes to a given prime’s square by fixing this method?

Like in the prime number theorem, but only with prime squares. It is an equation made out of the Sieve of Eratosthenes where x is the number of primes to the given prime’s square. Take a prime, in this case 7, and square it to 49. There are 3 primes…
Marci
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Is this generator of large consecutive prime numbers unique?

I've found no consecutive prime number generator on the internet that generates consecutive primes greater than about 17 digits. This program will generate primes of up to 290 digits: https://jprime.netlify.app And this YouTube video explains how…
Pat Lowery
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Are these prime k-tuplets, doublets, triplets, quadruplets, or?

I'm learning. Trying to figure out what this is a variation of. It's cool and I want to learn more about it. It's been fun to watch as my app runs into them. Any way, in a search for twin primes, I noticed that if I Take a positive number…
user1125455
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How to get a number into the form of $p^{s} \times n$?

In an article about the Miller-Rabin primality test, in the example section it says: "Suppose we wish to determine if $n = 221$ is prime. We write $n − 1 = 220$ as $2^{2}\times 55$, so that we have $s = 2$ and $d = 55$." My question is: What are…
Mazen Harake
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Which expression best represents the error in the prime number theorem?

From prime number theorem, we kmow that $$\pi(x) \sim \frac{x}{log(x)}$$ Fig 1.1 As you can see, in this graph, $\pi(x)-\frac{x}{log(x)}$ is increasing. My question is is there any function which best represents this curve?
Bjorn
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Infinite or Finite number of primes under a condition of arithmetic progressions

Infinite or Finite number of primes $p$ such that $p$ has a form: $p + 1 = \left( {\frac{{k - 1}}{2}} \right)\left( {\frac{{k + 1}}{6}} \right)$ where $k=5 \text{ } (mod \text{ }6)$ ?. Remark that since $k=5 \text{ }(\text{ }mod\text{ } 6)$,…
Nguyen Dang Son
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Probability of sum of two numbers being prime

Suppose we choose two random distinct numbers from $\{1,2,\cdots,n\}$. I learned that the approximate number of primes less than or equal to $n$ is given by $\frac{n}{ln(n)}$. Assuming that there are approximately $2n$ possible sums, the number of…
table
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Prime numbers as factors

A prime number greater than 5, say P, is a factor of N, where N is 1 repeated (P - 1) times. For e.g.: 13 is a factor of 111111111111(1 repeated 12 times, don't know a better notation) 23 is a factor of (22 times 1) This holds for larger numbers…
Sandeep Gupta
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identity of prime counting function

is this identity trivial ? $$ \pi (x) = \sum_{n=2}^{x} \frac{\Lambda (n) \mu (n)}{log(n)} $$ the mangoldt function is only nonzero for primes and prime powers and for any power of prime higher that 1 $ \mu(p^m) $ that'
Jose Garcia
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How to generate a large number that may or may not be a prime?

I wish to find an $n$ such that I can claim that it is unknown (with our current technology) that the following number is or isn't a prime. My initial starting point is this: $$ p=2^n-1 $$ where $ p \gg 2^{82,589,933}-1 $ which is the largest known…
Anon21
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Is $\forall y \, \forall z \, y \cdot z = x \rightarrow y = 1 \lor z = 1$ the correct way to describe a prime number?

The definition is taken from here. Is $$\forall y \; \forall z \; y \cdot z = x \rightarrow y = 1 \lor z = 1$$ the correct way to define a prime number? Do you have a link to an article where prime numbers are defined this way? If you would have to…
anon
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Are the three points (2,3), (11,13), and (47,53) unique?

The three points having consecutive prime co-ordinates (2,3), (11,13) and (47,53) are collinear. Are these the only three such points?
J. M. Bergot
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Optimal prime/semiprime/composite test not working for small numbers?

There is an algorithm to check is a number prime/semiprime/composite. Pseudo code: Input a number N and if N – 1 and N + 1 is not divisible by 6 then the number N is Not Prime. else it is prime or semi-prime If n-1 or n+1 is divisible by 6…
Kos
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Why can I get number of carries in such a strange method?

Here is the thing. I solved kata on Codewars with task: Find number of carries. I solved this task trivially. I reversed both numbers and sum each digit one by one. I accumulate sum / 10 (integer division). I increase counter each time when…
Ian Kirsh
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Simplify, expressing answers in terms of powers of primes?

So I'm not really sure how to express these two expressions in terms of powers of primes. Help? $16\cdot25\cdot5^2$ and $24^5$
figurine
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