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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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How to prove that if $c$ >$8/3$ then there exist a real number $\theta $ such that $\bigl\lfloor\theta^{c^n}\bigr\rfloor$ is prime

How to prove that if $c$ >$8/3$ then there exist a real number $\theta $ such that $\bigl\lfloor\theta^{c^n}\bigr\rfloor$ is prime for every positive integer $n$?
liesel
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An inequality on prime gaps

Is there a good inequality for prime gaps. Like $p_{k}-p_{k-1}\leq f(k)$ ? In other words is there a known upper bound for $p_{k}-p_{k-1}$?
esege
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A question about powers of prime numbers

Can someone help me in this problem? Let $p,q$ be prime numbers with $p < q$. There exists $m \in \mathbb{Z}^+$ for which $1+p+p^2+...+p^m$ is a power of $q$. There exists $n \in \mathbb{Z}^+$ for which $1+q+q^2+...+q^n$ is a power of $p$. Prove…
Jonathan
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Plotting Primes

This is a double question... I would like to see a plot of primes such that there are concentric circles, with each circle representing a prime and having its number represented as evenly placed points around the circle. For example, r=2: two…
bluedog
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Non-Magic Prime Number Greater Than $2^{256}$

I'm looking for non-magic prime numbers (ones that are very clearly not arbitrarily hand-picked) to use in a python library. Right now, I'm using mersenne primes, but I need one prime that is at least slightly larger than $2^{256}$ (at least $256$…
Ryan Shea
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Finding probable primes with large numbers of digits

I have a homework assignment to find the two smallest probable primes with $12$ digits, where a probable prime is defined as a number such that $a^{p-1} \equiv 1\ (\textrm{mod}\ p)$, where $a = 2, 3, 5, 7$. I'm obviously not asking for this to be…
Nate
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An interesting (unknown) property of prime numbers.

I don't know if this is the right place to ask this question. Please excuse my ignorance if it is not. I like to play with integers. I have been doing this since my childhood. I spend a lot of time looking up new integer sequences on OEIS. Last week…
Prometheus
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Help With Euclid's Proof that there are Infinite Primes

I need some help understanding a given proof so that I can in turn prove something. In a question I am given that: In order to prove that there is no "largest" prime number, for any prime $p$ just show there is a new prime $q$ such that $q>p$.…
user66807
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A prime is not a product of primes, so why is “every positive integer, except 1, […] a product of primes”?

I'm trying to understand why the fundamental theorem of arithmetic is phrased like this: Every positive integer, except 1, is a product of primes. A prime number is an integer but it is not a product of primes. It seems to me that "every positive…
zeynel
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A bound for a certain set of prime numbers.

Let $p_n$ denote the $n^\text{th}$ prime. Find a lower bound for $\left|S\right|$ where $$S = \left\{ q \in \mathbb{N} \mid q \text{ is prime and } p_n - n \leq q \leq p_n + n \right\}. $$ Any good bounds known? See this graph:…
Sophia Antipolis
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How quickly can we find a prime at least as great as $n$?

This may be trivial, but I'm wondering a few things. Is there an easy way to find a prime of the form $2k+1>n$ for some $n$? EDIT How quickly can we find a prime greater than a given number $n$?
Matt Groff
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Fraction of prime number between consecutive powers of two

A $n$-bit integer is an integer $x$ such that $2^{n-1} \le x < 2^n$. In [1] it is claimed without proof that a corollary of the prime number theorem is that: For any $n > 1$, the fraction of $n$-bit integers that are prime is at least…
Steven
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Why do we observe different color densities in columns and rows in different diagonal quadrants within the Ulam spiral?

There are 4 attached pictures of Ulam spirals. A very zoomed out white version of the Ulam spiral The same Ulam spiral but colored based on the x coordinates. The same colored version, but zoomed in with some numbers and the 0,0 origin coordinate…
Michael Tidwell
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Why $f$ is injective? (infinitude of primes)

In this very short paper by Dustin J. Mixon, I would like understand why the author says $f$ is injective by the fundamental theorem of arithmetic. In my opinion, the Fundamental Theorem of Arithmetic (FTA) is necessary to define $f$, but it…
Pedro
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For any integer a and c such that $a \gt c$, is it true that at least one integer in the range $[a, a+c]$ will have a prime factor greater than $c$

For any integer a and c such that $a \gt c$, is it true that at least one integer in the range $[a, a+c]$ will have a prime factor greater than $c$? If this happens to be true, what about if an integer in the range will have a prime factor greater…
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