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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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General form of a prime greater than 3.

Is the general form of a prime number $> 3$, $6a \pm 1$, or $3a \pm 1$. I've seen both used here.
Math1
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Does there is a mathematical proof of this fact?

The question is related to this page (https://en.wikipedia.org/wiki/Brun%27s_theorem) (in the section: Asymptotic bounds on twin primes) on twin primes. I am interested on the line where the author says: Brun's constant could be an irrational…
Safwane
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10001 and in general $10^k + 1$, how can we identify without brute force that such numbers are not prime?

I know that $137 \times 73 = 10001 $. I am looking for a properly reasoned approach. I believe that there is some general result also which says that all numbers of the form $10^k + 1$ (for $k>2$) are NOT prime. Why? I mean can the general…
Shraddheya Shendre
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Find natural numbers $n$ so that $n^n+1$ and $(2n)^{2n}+1$ are all primes

I have this question: Find natural numbers $n$ so that $n^n+1$ and $(2n)^{2n}+1$ are all primes. My idea is that we consider: $n=1 \Rightarrow n^n+1=2$ and $(2n)^{2n}+1=5$ (correct) $n=2 \Rightarrow n^n+1=5$ and $(2n)^{2n}+1=257$ (correct) And…
N.Paul
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Show that $q$ is prime

Let q be a positive integer such that $q \geq 2$ and such that for any integers $a$ and $b$, if $q|ab$ then $q|a$ or $q|b$.Show that $q$ is a prime number. It will probably be proved by contradiction, so I assumed $q$ is a prime, i.e $q=xy$…
Our
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Prime number problem

Find positive integers $x,y$ and prime $p$ so that $(xy^3)/(x+y)=p$
N.Paul
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3 Primes in a row?

Show $p$, $p+1$ and $p+3$ are all primes iff $p=2$ This is, of course, easy to prove one way. That is, assume $p=2$, then $p+1=3$ and $p+3=5$. However, I am not sure how to prove the reverse? That is, if $p, p+1, p+3$ are all primes, then $p=2$. How…
Math1
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Gruenberger's prime path

hi i'm looking for some historical informations about Gruenberger path. it is a path based on conjecture that every prime could be written in the form $6k+1$ and $6k-1$ (but what is the name of this conjecture?) i don't find a lot in the net.. the…
nkint
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Calculate exact number of primes up to N

How can I calculate exactly how many primes there are up to 65,025? I have seen that $x/logx$ gives an estimation, but it is not hugely accurate.
KOB
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Calculate the total number of equivalence classes

Let $P$ be the set of all primes. Define a relation in $P$ by $x\sim y$ if $x+y$ is even. It turns out that $\sim$ is an equivalence relation. Calculate the total number of equivalence classes. The only even prime is 2, so we can consider all the…
Rot Civ
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Infinitely many primes

Why is it that if I multiply a certain number of consecutive primes starting from $2$ and add $1$, I get another prime? This property is used it prove that there are infinitly many primes, but why is it correct?
Daniel Cortild
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sum of the largest and smallest prime factors

I got different answer from the answer sheet which says the answer is 14. What is the easiest way to find the largest prime factors of 5445? What is the sum of the largest and smallest prime factors of 5445?
learning
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Formula for coprimes between p and p# (primorial)

What I'm interested to find out about is if I can get a formula to find the sequence of coprimes for a primorial. ie for primorial 30 the coprimes: 1,7,11,13,17,19,23,29 EulerPhi(30) is the count of coprimes but I'd like to get the formula to find…
poker3
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Proof of a statement on prime numbers

Determine if the following statement is true or false. There exists a non-prime $n\in\Bbb N$ such that for all proper divisors $k$ of $n$ there exists $t\in\Bbb Z\setminus\{1,k\}$ with $t|k$. I proved that the statement is false by…
user142943
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Suppose $x$, $y$, and $z$ are pairwise coprime integers. Are $yz$ and $x(y+z)$ coprime?

Suppose $x$, $y$, and $z$ are pairwise coprime integers, in the sense that $gcd(x,y) = gcd(x,z) = gcd(y,z) = 1$. Are $yz$ and $x(y+z)$ coprime?
David
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