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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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Are there more numbers in this integer sequence? 3, 8, 12, 24

I was using some low primes for RSA encryption today, and noticed that for $p=5$, $q=7$, and ANY valid encryption key value, encrypt(encrypt(message)) = message. I thought this was an interesting property, so I found this sequence: $3, 8, 12, 24$.…
user125957
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T/F: The interval of width $n$ containing the most amount of primes is $[2,n+2]$?

Given $n\in\mathbb{N},$ the interval of width $n$ containing the most amount of primes is $[2,n+2]$ ( rather than $[2+x, n+2+x]$ ). This sounds like it should be true since the primes spread out more in general as $x$ increases. But general trends…
Adam Rubinson
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Do we know if there are more primes with even leading digits or odd leading digits?

I was just wondering, out of curiosity, do we know if there are more primes with even leading digits or odd leading digits? For example, primes with even leading digits would be $23$ or $29$ and primes with odd leading digits would be $31$ or $97$.…
Jeel Shah
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How to calculate prime numbers.

As a practice applciation, I am trying to write a prime number calculator that would be able to given a number, for example "124981242424", determine the nearest prime number and give me the ten next prime numbers in increasing order. I was trying…
LearnIT
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What is the simplest way to show (give an elementary explanation) on an intuitive level that $\sum_{ p \leq n}\frac{1}{p} \approx \log{\log{n}}$

What is a simplest way to get an idea (not rigorous proof) that $$\sum_{ p \leq x}\frac{1}{p} \approx \log{\log{x}}$$
Aleksey Druggist
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Is it possible that the formula for the $n^{th}$prime number is a polynomial in n?

I know that no pattern has been found yet. And prime numbers are weird, so the formula being a polynomial would be too simple to be true. Has there some proof been given that the expression for $n^{th}$ prime number can't be a polynomial in n? I…
user402662
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Last digits of primes

A prime number not equal to $2$ and $5$ can't have last digit equal to $2,4,5,6$ and $8$. Is it true that this is the only restriction on last digits of prime numbers? I mean if its true that for any sequence of digits with last digit not equal to…
user1006
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Why does $1$ divided by $p$ have $p-1$ repeating decimals?

Part of solving a bigger problem, I discovered that when dividing $1$ with a prime number $p$ > 11, the results has $p-1$ repeating decimals. Examples: $\dfrac 1{23} = 0.\underline{0434782608695652173913}0434782608695...$ As far as I could tell,…
Ole Drews Jensen
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use contradiction to prove that the square root of $p$ is irrational

On a practice exam, our teacher provides us with this question and this answer. Let $p$ be a prime number. Use contradiction to prove that $\sqrt{p}$ is irrational. ANSWER: By way of contradiction, assume $\sqrt{p}$ is rational. Then there exist…
Ali
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Proving that $(1+10^n)$cannot be a prime number when $(n>2)$

Proving that $$(1+10^n)$$ cannot be a prime number when $n>2$
E.H.E
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The sum of two prime number is $999$. What is their product?

Q. The sum of two prime number is $999$. What is their product? I know the answer is: $997+2=999$ so all you would have to do is multiply $997$ and $2$ My question is what if there was another question like this but the prime numbers were in the…
Caddy Heron
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Where is The third Gisella prime?

A Gisella prime is a prime number obtained from concatenating the first $n$ natural numbers starting from $2$ and then replace each composite on that concatenation with the concatenation of its prime factors in ascending order, i.e: $23456789101112$…
Gisella M
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a practical prime counting function

Looking at Prime counting functions on Wikipedia, I only found formulas with no hint on how people got there. So, to better understand, I've decided to build one from scratch, starting from a naive kind of Sieves of Eratosthenes. Here is what I've…
Eddy Khemiri
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Highest ratio between consecutive prime numbers

Let $r = p_2/p_1$; where $p_1$, $p_2$ are consecutive prime numbers. What is the highest possible value of $r?$ Are there any consecutive prime numbers such that $r > 5/3$?
mridul
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Generalisations of primes

I've read of (normal) primes, Gaussian primes and Eisenstein primes, which all uses different ways to define an integer to be a prime. For instance, $2$ factors into $1-i$ and $1+i$ for guassian primes. Although I'm thinking this might only be the…
Frank Vel
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