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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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Proof involving induction and primes

I'm looking to prove that: $$p_n \leq 2^{2^{n-1}}$$ Where $p_n$ denotes the $n$th prime in ascending order. The proof method is induction. I've solved my base case, that is: $n=1$ $p_1 = 2$, $2^{2^0}=2$, $2\leq2$ Therefore $P(1)$ is true. And the…
RJJ
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A quick way to determine whether a number is prime by hand?

I would like to know how to determine by hand whether a number is prime to help me in math contests.
zscoder
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show this number is not a prime number

The following problem is a special case Show this number always is composite number? , but I think this special case is relatively difficult to deal with, that is, if this is solved, it may solve the general situation. show this number …
math110
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Annoying primes

first of all, I'm no mathematician at all. I was just playing with prime numbers and ended up with this list: 2 = 2¹ 3 = 3¹ 5 = 5¹ 7 = 3¹ + 2² 11 = 2¹ + 3² 13 = 13¹ 17 = 5¹ + 2² + 2³ 19 = 2¹ + 3² + 2³ 23 = 11¹ + 2² + 2³ 29 = 17¹ + 2² + 2³ 31 = 19¹ +…
Alexandre Paloschi Horta
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Finding the largest prime in sequence 10101...

If you consider the (decimal) numbers 10, 101, 1010, 10101... where the 1 and 0 alternate, what is the largest prime number in the sequence? Thanks in advance for your help, I appreciate it.
Three OneFour
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Is every positive nonprime number at equal distance between two prime numbers?

For example $8$ is in the middle of the interval between $5$ and $11$, $9$ is at equal distance between $7$ and $11$; $10$ between $7$ and $13$.
Vlad Dimitriu
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Why does "the probability of a random natural number being prime" make no sense?

I read an essay about prime numbers. In it the author suggests that a natural first question to ask is, What is the probability of a random natural number being prime? but proceeds to dismiss it as "not making sense". I wonder what's wrong with…
Adam
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How to find number of prime numbers between two integers

I have two integers, x and y so that x < y. How many prime numbers are there between x and y (exclusive). Is there a formula or algorithm to compute?
Amp Tanawat
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Prove that there is no positive integer $n$ such that $1^{2000} + 2^{2000} + \ldots + n^{2000}$ is prime.

Prove that there is no positive integer $n$ such that the following number is prime: $$S_n = 1^{2000} + 2^{2000} + \ldots + n^{2000}$$ I was thinking about the last digit of the number. For certain values of $n$, the last digit of $S$ is even, so…
gareth618
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What is the largest $n$ for which the $n$th prime is known?

$p = 2^{43,112,609} - 1$ is currently the largest known prime, but the $n$ for which this $p$ is the $n$th prime is, presumably, unknown. What is the largest $n$ for which the $n$th prime is known? (For the sake of definiteness, let's say a number…
r.e.s.
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Prime numbers of the form $(1\times11\times111\times1111\times...)-(1+11+111+1111+...)$

Let $$R(1) = 1-1,$$ $$R(2) = (1\times11) - (1+11),$$ $$R(3) = (1\times11\times111) - (1+11+111),$$ and so on... $$R(4)=1355297\quad\text{(a prime number!)}$$ $R(4)$ is the only prime I found of such form up to $R(200)$. Are there anymore primes of…
Henry R
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Will everyone skip a prime in this game?

Suppose there are n people sitting at a round table. The game begins with the first person saying the number $1$. Then the game proceeds as follows: Players take turns going round the table. If the next number is composite, say that number and your…
Mikołaj Bilski
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Does $(10^{6n+5}-54n-46)/162$ ever be a prime for $n>0$?

$(10^{6n+5}-54n-46)/162$ is prime for $n=0$, but after that I didn't find anymore prime up to $n=500$, I didn't even find a single semiprime with that form (!!). Is there prime number of such form for $n>0$ ?
Kazuki NR
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How do I prove these three statements true/false?

I am still being taught maths at school, so all these fancy symbols like $\sum$ I have no idea about (although writing it out through mathjax, I think it possibly means the "sum" of something) but I believe to have made three potential discoveries…
Mr Pie
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Why is it that the product of first N prime numbers + 1 another prime?

Recently I came across this proof for fact that primes are infinite. It's a proof by contradiction. The proof assumes that primes are finite and there is a prime M which is larger than any prime out there. Then you basically take the product of all…
user2340452
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