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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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Prove or Disprove: There are infinitely many integers $n$ such that the three integers $n$, $n+2$, $n+4$ are all prime.

Prove or Disprove: There are infinitely many integers $n$ such that the three integers $n$, $n+2$,$n+4$ are all prime. (Suggestion: Try some sample values of $n$ and look for a pattern.)
klorzan
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For $x < y$ and $x$ and $y$ are coprime, is it true that $y-x$ and $y$ are coprime?

I'm working Problem 72 on Project Euler and had this thought that if $x/y$ is a proper fraction then naturally $(y-x)/y$ is as well such as 3/8 and 5/8 or 4/15 and 11/15. Growing up I've always noticed this symmetry but never have I questioned it…
Corey Ogburn
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If $x$ and $y$ are prime and $y^2+y$ divides $x^2+x$ then $\frac12(x-y)$ is composite

Assume that $x\ne y$ are prime and that $y^2+y$ divides $x^2+x$. Prove that $\frac12(x-y)$ is composite. My approach: Let's write $x^2+x$ and $y^2+y$ in a different form: $x(x+1)$, $y(y+1)$. We can now notice, that since $x$ (and also $y$) is…
TomDavies92
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Mr. Norata's conjecture, semiprime of the form $16^p+1$, with p is a prime

I have already posted this question with username "mamihlapinatapai", but apparently I've made a major/big error to my question. I couldn't edit it because my question has been answered by two users. My previous statement that $16^n+1$ can be a…
Mika Setiawan
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Name for prime numbers with only prime digits?

I'm wondering, is there a name for a prime number where all digits are also prime? Some examples: 37, 53, 3253, 5573, 23753. I've been calling them 'double primes', but I doubt that's the correct term (if there is any).
Anders Sandvig
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Does $1+x+x^2 ... x^{p -1}$ being a prime number imply p is prime?

Let $p$ and $x$ be two positive integers greater than 2. If it is given that the sum : $$1+x+x^2+x^3... x^{p -1}$$ is a prime, is it possible to prove or disprove that $p$ is prime? If so, what would this proof be?
praeseo
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Prime counting function

How much of an impact would the discovery of an exact formula that is equivalent to the prime counting function have on the mathematics community and acedemia as a whole?
Wilson22
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Is $u_n$ where $\left\{\begin{matrix}u_{1}=5 \\ u_{n}=\frac{2^{u_{n-1}}+1}{3} \end{matrix}\right.$ always prime?

$\left\{\begin{matrix}u_{1}=5 \\ u_{n}=\frac{2^{u_{n-1}}+1}{3} \end{matrix}\right.$ I conjecture that $u_{n}$ is prime number. But I can not prove it. So I want to know my conjecture is right or wrong.
Mạnh Nguyên Nguyễn Hoàng
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Prime numbers of the form $P^Q+R^S$

Is there a prime number of the form $P^Q+R^S$ where $P,Q,R,S$ are four distinct prime numbers? Examples: $2^3+7^5$, $2^3+5^{11}$ are not primes, $2^5+11^7$ is not a prime.
karra zanny
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Will the Product of a Set of Primes above 1 ever be Equal to The Product of a Different Set of Primes above 1

Given two sets a and b. When a and b only contain primes above 1, will the product of every number in a ever be equivalent to the product of every number in b?
champfish
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If $N\equiv 1\pmod 4$ does then follow that $p\equiv q\equiv 1\pmod 4$

$N = pq$ is the product of two primes. If $N\equiv 1\pmod 4$, does then follow that $p\equiv q\equiv 1\pmod 4$ ?
testboy
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Why is $0$ the $0^\text{th}$ prime?

I found this question on $\prod_{n\to \infty}(1-1/p_n)$, played a little at Wolfram's Alpha and found the following: The series expansion of a related indefinite integral $\int \log (1-1/p_n)dn$ gave a series expansion at $n=0$ of $$ n…
draks ...
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A relationship among the first $n+1$ primes

Consider the set $P_{n+1} = \{p_1, \dotsc, p_{n+1}\}$ of the first $n+1$ primes. Does there always exist a $p \in P_{n+1}$ and a partition $\{A, B\}$ of $P_{n+1} \setminus \{p\}$ (in other words, $A$ and $B$ are disjoint and nonempty subsets whose…
Mario Gomez
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Finding prime numbers in 3 equations

I have to find prime numbers P1, P2, P3 and P4 that satisfy the 3 equations below: P2 = P1 + 2 P3 = P2 + 4 P4 = P3 + 8 And I'm clueless about where to start. Which mathematical theorem/method (if any) that I could use to aid me with this question?
apple
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Equation involving prime sequence

I tried to solve this equation: $$3^{5+7+11+13+...+P(k-3)+P(k-2)}=P(k-1) \mod [ P(k) ],$$ where P(k) here is the $k^{th}$ odd prime number. The only solution I have found is $k=52$ or $P(k)=241$. Could you find the next solutions for $k$?
Tanaka C
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