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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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Does there exist a prime that is only consecutive digits starting from 1?

This is a problem I came up with the other day, and have absolutely no clue how to solve. The problem is: does there exist a number in the set $K$ that is prime, where $K$ is defined to be the set of all numbers that follow this…
ASKASK
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Sum of Prime Factors - TopCoder

Recently in Topcoder, I faced a problem which stated as follows: "You have a text document, with a single character already written in it. You are allowed to perform just two operations - copy the entire text (counted as 1 step), or paste whatever…
StrictMath
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Are all prime numbers up to Mersenne Prime 48 known?

In January this year the biggest prime number so far has been found and it is a Mersenne Prime. $$2^{57885161}-1$$. My question: Are all prime numbers from 0 up to $2^{57885161}-1$ found, or not? If you only look for prime numbers which are Mersenne…
user94342
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An alleged formula for primality test

I stumbled on a claim on the internet the other day, which alleged that the following formula tests the primality for integers…
avocado
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Largest prime representable in javascript

In Javascript, the largest integer that can be represented exactly is Number.MAX_SAFE_INTEGER, with a value of $ 2^{53} - 1$. What is the largest prime value that fits under this value threshold? I cannot find a suitable reference for this value on…
G987
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Is a function useful that has zeros on the prime integers?

So I want a function that is zero on the reals only on the prime integers and which doesn't depend on knowing the primes. I construct: $$f(x) = e^{-x^2} - \sum\limits_{n=2}^\infty e^{-n^2} \frac{ \sin(\pi x)^2 }{ n^2\sin(\pi x/n)^2}$$ Which has…
zooby
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For prime $p$ do we have $p^3+p^2+p+1=n^2$ infinitely often?

This is a question to ponder about the occurrence of prime $p$ giving $p^3 + p^2 + p +1=n^2$ as is true with $7$ giving $400=20^2$. Do you think this will ever happen again?
J. M. Bergot
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Prime sum identity

Let $ \Lambda(k) $ denote the von Mangoldt function: $$ \Lambda(k) \stackrel{\text{def}}{=} \begin{cases} 0 & \text{if $ k $ is not a prime power}, \\ \ln(p) & \text{if $ k = p^{j} $}. \end{cases} $$ Also, let $ \lfloor x \rfloor $ be the floor…
Ethan Splaver
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Pair of prime numbers to have the same product

Say we have 2 prime numbers $a$,$b$ and $c=a*b$. Is there any other pair of prime numbers $x,y$ (distinct from $a$ and $b$) so that $c=x*y$ ?
Beach Williams
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Is there a prime number less than the product of consecutive primes, but greater than the last consecutive prime?

Let $P(k)$ be the product of $k$ consecutive primes $p_1, p_2, \dots, p_k$. So, e.g. $P(4)$ is $2 \cdot 3 \cdot 5 \cdot 7 = 210$. Is anything known about whether $P(k) > p_{k+1}$ is always true (for $k > 1$)? It seems like it should be true, since…
Kyle Kaitan
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Prove or disprove $p_1p_2\cdots p_n+1$ is prime for all $ n\geq 1$

Let $p_1=2$, $p_2=3$, $p_3=5$ and, in general, let $p_i$ be the $i$-th prime. Prove or disprove that $$p_1p_2 \cdots p_n+1$$ is prime for all $ n\geq 1$ Well I was able to find a counter example when $n=6$ but I do not have a general way to show why…
HighSchool15
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What is a covering set of a Sierpinski number? What does it do?

Recently a new prime number has been discovered, which eliminates one of the six remaining candidates for the smallest Sierpinski numbers. So I was reading the wikipedia article about the Sierpinski number, where I came across what is called a…
Kushal Bhuyan
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the least odd prime leaving prime remainder when divided by the first $n$ odd primes

What is the least odd prime giving a prime remainder when divided by $3$? It is $5$. What is the least odd prime giving prime remainders for both $3$ and $5$? It is $17$. For division by $3,5,7$ it is also $17$. For division by $3,5,7,11$ it is…
J. M. Bergot
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What is the smallest composite number

I got the correct answer for the following problem by trying all numbers. It's very time consuming. Can anyone tell me whether there is a simple and easy way to solve the problem? What is the smallest composite number generated by $p^2-p-1$ where p…
learning
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find a 5-digits $q$ such that $2^q+17$ be a prime

How we can find a 5-digits number $q$ such that $2^q+17$ be a prime number? Does there exist such number?
SKMohammadi
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