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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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Proof only by contradiction

I know that my question might be ambiguous and unclear . Question : Prove by contradiction that if $p$ is a prime number then it can be written in $p = Sk + 1$ or $p = Sk + 5$ forms . I don't know what is the meaning of $S$ . If someone explain to…
S.H.W
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How to know if a number 10^n + x is prime?

Is there a rather quick/easy way to check if a number in the form 10^n +x is prime or not, given that n and x are known. I should also not that for our case n is mind-blowingly huge, and x is rather small, in magnitude of hundred.
TheDeltak
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Is zero infinitely composite?

If I divide zero by any number, I get zero. Would that not make it composite, perhaps even infinitely composite? Edit: I AM NOT ASKING IF IT IS PRIME. I AM ASKING IF IT IS INFINITELY COMPOSITE
user406613
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What is prime for other hyperoperator?

I want to appologize for lack of therms, because I just love math as hobby, not on professional level. We all know what prime number can't be split (by division) on two different numbers aside from 1 and itself. So, basicaly, this numbers is…
eocron
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Show that $17$ is the only prime of the form $p^q +q^ p$ , where $p$ and $q$ are prime

Show that $17$ is the only prime of the form $p^q +q^ p$ , where $p$ and $q$ are prime My attempt so far is first assume $p$ and $q$ are prime. Now $17=2^3+3^2.$ Now fix $p=2$ and let $q>3$ then $q=3x+1$ or $q=3x+2$, $x \in \mathbb{Z}$ then…
HighSchool15
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How to shorten prime numbers in database?

I'm bad at Math and english isn't my native language. Bear with me. Thanks. I'm running a prime number search script and writing the result to a SQLite3 database. Now I'm looking for a way to shorten these prime numbers because I don't want to be…
Ninja Cat
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Triple Prime Proof

Note: Someone else has posted this proof as a question; however, that post is largely inactive. I am interested in exploring this proof, but don't know how to "bump" someone's post, so I'm re-posting it here. If there is another way I should go…
Math1
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Uniqueness of the sum of $n$ prime numbers in a range

Let's say that I want to calculate the sum of 20 prime numbers that reside in the range up to 1000. Will the sum (addition, not multiplication) of any 20 primes in that range be unique , or will I see recurrences?
azer
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What, intuitively, are supernatural numbers?

Before I can get to grips with something rigorously I need to understand it in an intuitive way. I was trying to get my head around the supernatural numbers. Looking at the definition it looks like they could extend the conventional numbers, by…
it's a hire car baby
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Euclid Mullin Sequence

Consider the Sequence as follows. Let $a_1 = 2$, $a_n$ be the largest prime divisor of $P_n = 1 + {\prod_{i = 1}^{n - 1} a_{i}} $ Then we obtain a sequence of prime numbers How do you show that 5 is never in the sequence? OK, I am not quite…
Phantom
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Finding the last digit of largest mersenne prime

I was wondering what the best way was to find the last digit of the largest known Mersenne prime $2^{6972593} - 1$? Is there any logical way to do this or will I have to just some how compute the answer and then find the last digit?
Jeel Shah
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A Formula For Primes

Could someone explain me why this arithmetic of sets can not be called a Prime Numbers formula? Was it already found before and is not relevant? Prime numbers sequence $\mathbb P$ (or set of members expressed as a sequence) is revealed from a…
T.Kalnius
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With which natural value of n, the polynomial will be prime value and why?

So. $P(n) = n^4 + n^2 + 1$ is a polynomial. I calculated that answer is 1. But I don't understand why?
sashaaero
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Sum of reciprocals of n-digit primes

I have observed, by calculation, that the sum of the reciprocals of all the n-digit prime numbers is approximately 1/n, and that this becomes increasingly accurate as n increases. Is there a simple way to prove this? I have also observed that this…
John Grint
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