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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 infinitely many primes $p_n$ for which $p_n< 2n$?

I am working in some calculations and I found this kind of problem. That is, are there infinitely many primes $p_n$ for which $p_n< 2n$ ? I think the answer is no, but I dont have a formal explanation.
GoldSoundz
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Does there exist prime number of the form $n^n+s(n)^{s(n)}$ after the trivial one $1^1+1^1$?

Let n be a positive integer and let s(n) be the digit's sum of n. Is there a prime number of the form $n^n+s(n)^{s(n)}$ ? after the trivial one $1^1+1^1$ ?, I've checked n up to $2500$ without finding a prime anymore (!). I found that numbers of…
Johnny G
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Exponential running time for factors

I'm trying to read Sipser's book chapter 10 on page 399 and there's a paragraph that says One way to determine whether a number is prime is to try all possible integers less than that number and see whether any are divisors, also called factors.…
testinggnitset ser
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Determine when $(p-1)/(k+1)$ is integer for prime $p$ and $k \in \left\{{0,1,\cdots,\left\lfloor{(1/2)(3-\sqrt{5})\, p}\right\rfloor-1}\right\}$

Obviously this holds for $k = 0$ and $k = 1$ sine $p - 1$ is even. The limit on $k$ is about $0.38 p$. This limit is due to the problem details however a general solution if possible for $k \le p - 2$ is just as good.
Lorenz H Menke
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Find number of prime numbers between two big numbers

I am given two numbers $n,x$. I have to find the number of primes between them. Difference can be up to $10^6$; thus looping will take too much time. Is there any other way I can do it?
Murad
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How to find out if there 7 prime subtractors of a number

I have a number and i want to find out if i can get that number by adding 7 prime numbers.How can i find that out? Example Number:14 Answer:2 2 2 2 2 2 2
Murad
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Uniqueness of numbers composed of primes

TL;DR: If X is a number that is created by multiplying n unique primes together, is it unique among all similarly created numbers? My motivation for this is: Let's say I have an array of 20,000 unique strings ordered alphabetically (not important…
argyle
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Find all positive integers x and y such that x, y, x + y and x − y all are primes

Find all positive integers x and y such that x, y, x + y and x − y all are primes. Ohkay now i have already solved the problem but i am looking for other methods to solve it. Here is my work out Since x − y is a prime, x − y > 0 =⇒ x > y. Suppose…
Pole_Star
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infinitude of primes with primitive pythagorean triple variables

I just want to share what I found, I don't know if this is something useful or worth knowing: Let $(x,y,z)$ be a primitive pythagorean triple, odd $y$, then there are infinitely many primes of the form: $(x^3 + y^3 + z^3)/(z+x)(z+y) - (z-y)/2$
unknownMe
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Create unique identifier for 7 cards from 52

This is for a poker lookup In this case a hand is 7 cards (it is really the best 5 but can ignore that) In a deck of 52 have 133,784,560 7 card hands - the order of the cards does not matter Cards can be identified as 0-51 or 1-52 For each hand I…
paparazzo
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Divide a composite number

Any hint for this demonstration? Show that if $\displaystyle n$ is a composite (positive integer) number it has a divider that is less or equal to $\displaystyle \sqrt {n}$.
Oualid
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Prime number interference patterns

Please do excuse this observation, as I am not a mathamatician - just a Scottish Lawyer who (for some mad reason) loves fractals and prime numbers. Why is it, when I play around with simple images of Ulams Spiral at high resolution on my IPad, I…
Donna Skelly
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Prime number problem

Find natural numbers $x$,$y$ and prime $p$ so that $y^4+4=p^x$. In my opinion, i think we should consider cases y=0 and y=1. Hence (x;y;p)=(2;0;2), (1;1;5). Then we proof: if y>1 =>there's no solution.
N.Paul
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6h element of iteration under the function $f(x) = 2x + 1$

So I was watching this numberphile video and it explained how the iteration under the function $f(x) = 2x + 1$, starting with $x_1=1$, (the mersenne sequence) always produced a number with one or more prime divisors not seen before. EXCEPT 63. Dr…
Brian010515
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How to find all prime numbers in the form $n^m+1$ or $n^m-1$?

I was asked to find all the prime numbers in these forms and prove that these are the only prime numbers in these forms. From some basic research on the internet, I suspect it may have been a trick question and that it is conjectured that they are…
John Smith
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