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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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Is there a database of prime numbers of the form ...

I was hoping to find an exhaustive list or database of the most common forms of prime numbers that is searchable. By form, I mean 2^p-1 or 6n-1 etcetera. Does anyone know of a good resource? The closest I have come to such a resource is oeis.org.
niobe
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Twin-prime averages

Sorry if this is trivial! Consider the set 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, These are all such that $2x-1$ and $2x+1$ are both prime. Why are they all divisible by 3? And if say $yx-1$ and $yx+1$ are both prime (i.e. generated by the…
user42520
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Show that if $p=2^a -1 $ is prime then a is prime

Show that if $p=2^a -1$ is prime then $a$ is prime. I found this problem very hard to solve and I only managed to show that $a$ must be odd; if $a$ was even we could rewrite $p$ as $$p=2^{2k}-1=(2^k-1)(2^k+1)$$ with $k$ different than $2$. Other…
PunkZebra
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Infinite generation of primes

I understand the prime sieves find primes less than a value, say n. I also know that trial division is woefully inefficient for any sort of large number. Is there any way of efficiently finding primes, without the arbitrary limit that a sieve needs…
MagikCow
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What is the extent of the role of prime numbers in mathematics?

I understand that prime numbers are called the "atoms" of number theory, because of the fundamental theorem of arithmetic. I also understand that they have some important practical applications such as in cryptography. What I am wondering is, how…
user56834
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What is the simplest to prove, non-trivial upper bound of prime counting function?

The thread here lists some pretty unique, simple, and non-trivial proofs on lower bounds for the prime counting function. We can also ask the analogous question: What is the simplest to prove, non-trivial upper bound for the prime counting…
Sandeep Silwal
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Prime ratio limit

I'm curious about approximating numbers as the ratio of primes. let $p_n$ be the nth prime let $g_r(p)$ be the smallest prime greater than $r \times p$ looks like $\lim \limits_{n \to \infty} g_r(p_n)/p_n = r$ Is this true? How to prove it?
Ken
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Is the set of dyadic rationals a field?

I recently learned that the dyadic rationals is the set of rational numbers of the form $$\frac{p}{2^q}$$ where $p$ is an integer and $q$ is greater than or equal to zero. I think the set of dyadic rationals is not a field. Here's why: One of the…
Roger Costello
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Product of its Prime Factors

Given that $4095 = 8^4 - 1$ write $4095$ as a product of its prime factors. I know how I could separate $4095$ into prime factors however I'm not sure how I could use $8^4 - 1$ to help me. I could perhaps move the $1$ over to get $4096$ and then…
itshanks
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asymtotic ratio of nonsquarefree repunits

Let $R_n:=\frac{10^n-1}{10-1}$ (called a repunit) and $\mu$ be the Moebius function. Also $[n]:=\{1,2,3,\cdots, n\}, A_n:=\{m \in [n]| \mu (R_m)=0\}.$ What is the value of $\lim \limits_{n \rightarrow \infty} \frac{|A_n|}{n}?$ Find the values of…
hkju
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What does Terence Tao mean by the statement "primes behave randomly"?

http://164.67.141.39:8080/ramgen/specialevents/math/tao/tao-20070117.smil The Riemann hypothesis is, according to Tao, equivalent to the idea that the primes do behave randomly -- they are distributed according to the prime number theorem, with an…
D J Sims
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What is the probability of being prime?

Given some arbitrarily large $a$, what is the probability that this number is a prime number? My attempt involves seeing that for $a$ to be prime, then it must not have a factor $N$ in the following range: $$2\le N\le\sqrt{a}$$ Which helps make the…
Simply Beautiful Art
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What's the smallest number that we don't know if it's prime or composite?

What's the biggest $n$ such that for all $1
Filip Haglund
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Does the sum of digits of prime number written in base7 never divide by 3?

I have tested all the primes up to 50,000,000 and did not find a single prime which satisfies the condition "sum of digits of prime number written in base7 divides by 3". E.g. 13 (Base10) = 16 (Base7) --> 7 (sum of digits in base 7) 1021 (Base10)…
Buchas
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The last p digits of $p^p$ form a prime number,where p is a prime

Let p be a prime number, here I'm interested with the last p digits of $p^p$ if it forms a prime number. And if I've not mistaken, the smallest p with that property is $433$. Meaning that the last $433$ digits of $433^{433}$ is a prime number. I…
Takuya Nazuga
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