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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.

12562 questions
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Is this estimation correct and how I can measure the corresponding error

We know that the number of primes in a given interval $(0,x)$ is about $x/logx$. Now, I have an inequality of the form $p_{n}≤x$ where $p_{n}$ is the $n$-th prime. Then I deduce that $n
Safwane
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Determine whether 177 is a prime.

The question is pretty straight forward... Determine if 177 is prime using the prime number theory. I am confused what type of answer to give. The question doesn't seem to be asking for you to explore all $n < \sqrt{177}$ and check the gcd. This is…
CarterKF
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Why is that for $x = \frac{\sum_{n=1}^{(p-1)} n}{p}$, where $p$ is a prime number, $x$ is always a positive integer?

Here is a graph for the first few primes vs sum. prime vs sum
mnkystack
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Wäre es ein Fortschritt, die Anzahl der Primzahlen (Menge) bis zu einer gewählten Größe zu berechnen oder ist es bereits bekannt?

Leider weiß ich nicht, wie ich hier eine Exceltabelle mit den Formeln und der Primzahlrechnung hochladen kann, bzw. ob es möglich ist. Ich bin sehr dankbar für Antworten! English Translation (via Google Translate): Title: Would it be an improvement…
Robert Finke
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About primes in an interval

I find this result without a proof: The prime number theorem implies that for any $\epsilon > 0$, there exists an integer $N$ such that for all $x > N$, there exists a prime between $x$ and $x(1 + \epsilon)$. I am asking about a simple proof for…
Safwane
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How to use Lehmer's formula?

Lehmer's formula for calculating the prime-counting function π(x) is given at http://mathworld.wolfram.com/LehmersFormula.html, but I can't figure out how to use it properly. Calculating π(3) results in a = π(1.32) = 0, b = π(1.73) = 0 and c =…
Måns Nilsson
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Is this can be an integer $\sqrt[2z-1]{2mz+p_{n}}$?

let the integers $z>2$, $m>1$ and a prime number $p_{n}\geq 3$. is the following can be an integer for any value of $z,m$ and $p_{n}$ ? $\sqrt[2z-1]{2mz+p_{n}}$
Shopify lover
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How many twin primes are there whose sum is a power of a prime?

How many twin primes are there whose sum is a power of a prime? I have started with the form p+(2+p), which reduces to 2(p+1), which suggests the number must be even. Where could I go from here?
ROS
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About finding a real $w$ such that if $k≥w$ then $p_{k}≥c$

Let $p_{k}$ be the sequence of prime numbers. Assume that we have an inequality of the form $p_{k}≥c$ where $c$ is real number. My question is about finding a real $w$ such that if $k≥w$ then $p_{k}≥c$ I am aware about the fact that the prime…
Safwane
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Square any prime p and see whether there are other numbers squared that have the same last digits.

If I square 37 I get 1369. Excluding any solution having 37 as the last digits, can I find other numbers when squared to give me 1369 as their last digits? Solutions are any numbers having last four digits 1213, 3787, 6213, and 8787. Do all…
J. M. Bergot
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Prime numbers relation

Show that if $k$ is a natural number. And $k$ is like $4*m +3, so $k$ has a prime factor in this form. Please, show me different ways to solve this question.
Santiago
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area of n-gon of prime points

Using prime points (2,3), (5,7) (11,13)....to a last prime point to form an irregular n-gon by connecting each consecutive point with a line and connecting the first point to the last point with NO previous line being intersected by this longest…
J. M. Bergot
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Primes having 2 as a primitive root

Let $p$ a prime and $a$ a quadratic residue of $p$. If there exists some positive integer $y$ satisfying $a\equiv2^y \pmod{p}$, prove or disprove that $2$ is a primitive root of $p$. My progress is that if 2 is not a primitive root then the set…
Hypernova
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Why are primes of the form $6k+1, 6k-1$ where the prime is $\geq 3$

I recently came to know that primes are of form $6k+1,6k-1$ for primes greater than three. Why is this so? I tried my hand on it could not really understand about it. I have also heard of Dirichlet's theorem but can there be any elementary such way…
user607476
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Find all primes $p$ for which $19 p - 1$ is a perfect cube

I set the equation to $19 p - 1 = n^3$, then got $19 p =(n+1)(n^2-n+1)$. I don't know what to do now.
Monnsom
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