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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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prime number greater than 100

I 'm confused about prime number. It is possible that we can find a not prime number that is greater than 100 and not divided by {2,3,5,7,9}. because someone said to me that we can check if a number(greater than 100) is prime just to check if it…
user3521250
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Function generating primes.

Is there any non-identity monotonically increasing one-one univariate function that takes prime number as input and generates prime number as output ? The asymptotic complexity to calculate output must me $O(1)$ (assume exponentiation is $O(1)$…
hanugm
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Primes expressible as $\frac{4k^3}{(k+m)^3}+\frac{6k^2}{(k+m)^2}+\frac{4k}{(k+m)}+1$ for $k,m\geq1$

Does anyone know how to express two primes such that $$P=\frac{4k^3}{(k+m)^3}+\frac{6k^2}{(k+m)^2}+\frac{4k}{(k+m)}+1,$$ where all numbers are nonzero integers?
Vassilis Parassidis
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All primes that cannot be be represented as a specific sum

For $n \in N$, $p_i$ prime, for $i \in N$, find all primes such that can be represented as $p_1 p_2\cdots p_n+p_1 p_2\cdots p_{n-1}+p_1 p_2\cdots p_{n-2}+\cdots+ p_1 p_2 + p_1 + 1$. Source: http://mishabucko.wordpress.com
tesgoe
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The set of prime numbers whose first digit are 1

Why cannot I use Dirichlet's theorem on primes in arithmetic progressions to compute the density of the set of prime whose first digit is 1? Thank you very much :)
user110706
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prove that the equation $4x^2 + y^2 = 4f^2 + 1$ has a unique solution y = 1 and x = $\sqrt{f^2} $ if $4f^2 + 1$ is prime.

prove that the equation $4x^2 + y^2 = 4f^2 + 1$ has a unique solution y = 1 and x = $\sqrt{f} $ if $4f^2 + 1$ is prime. I have no ideas, maybe it's impossible or the statement is false. I found proof thah equation $x^2 + y^2 = p$ has at least 1…
morz1k3
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Prove that if 11...11 (n 1's) is prime then n is prime

Prove that if $\underbrace{11...11}_{n}$ is a prime number, also $n$ is a prime number. There's already a similar thread here, but because of this weird requirement about having at least 50 reputation, I can't add a comment to ask for…
Konrad
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More information about Moser formula for primes: $ p_n=\gamma \cdot 10^{n(n+1) / 2}-10^n \gamma \cdot 10^{n(n-1) / 2} $

You will find the above formula here. The problem is I can't find any deduction of the formula above anywhere. I couldn't find the reference number 61 "Leo Moser, A prime representing function, Mathematics Magazine 23:3 (1950), pp. 163–164" on the…
R. S.
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"How to Prove It": using a product of primes plus $1$ to find a new prime

In the Introduction of the third edition of How to Prove It, Velleman presents the modern version of Euclid's proof that there are infinitely many primes, the one in which you take a number $m = p_1 p_2 \cdots p_n + 1$ and note that $m$ is not…
Kelly
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Checking prime numbers using sum og digits?

I have looked in some of my old books and found an exericse that I do not know how to solve. It seems pretty simple though. The question is as follows: Which of these integers are…
Grazosi
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Sorted Multiplicative Inverse Pairs (Plotted)

Does symmetry for (M2-M1) hold for all $P$, where $P$ is prime? Please read the info-graphic for clarification. Right click on the image and "open in a new tab". I cannot find a proof showing that for all $P$, their sorted multiplicative inverse…
user1007341
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Showing that no two primes subtracted can give 97

I want to prove that the it is not possible that when two prime numbers are subtracted, for them to result in 97: $$p-q=97$$ But honestly, I don't know how to go about it. Any suggestions?
Nick Heumann
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Sorting out prime and composite numbers

this is a question I was asked to solve: The mathematician wrote a three-digit whole number on the whiteboard and asked the students to decide whether that is a prime number or not. Big-brain Billy divided it from 2 to 31 but didn’t find any…
Leo
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Find all primes such that $p$ and $p^2$ are of the form $2k^2-1$

Find all prime numbers such that $\dfrac{p+1}{2}$ and $\dfrac{p^2+1}{2}$ are both squares of an integer. What makes it slightly annoying is that it is not just prooving that there is none, because 7 is such a prime.
Kekule
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Determine all prime numbers that can be written as

I have a question in this lesson: "Determine all prime numbers that can be written as $ n^{2} - 1 $, for some $ n \in N $" I tried to make $ n^{2} -1 = (n + 1) (n-1)$, but I didn't get any $n \in N $. Can anyone help me with this? Thanks.
Marina
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