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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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A sufficient condition to be a prime number (conjecture).

Let $p$ denote an odd prime power (a prime power is a number of the form $p^k$ where $p$ is a prime and $k \ge 1$). These are the numbers OEIS A061345 without the $1$. For $p \ge 5$ defined prev$(p)$ to be the (strict) predecessor of $p$ in the set…
Peter Luschny
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There's a simple proof that there's no limit on the size of prime gaps. Is there a simple proof that every possible size is represented?

This is similar to a question that has been asked but I am asking for something slightly different. It's been asked "are there prime gaps of every (even) size?" My question is, is there a simple proof that there are prime gaps of every (even) size?…
user3752935
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How many safe primes are there?

Compared to the number of integers and the number of primes, how many safe primes are there? Specifically, I'd like to be able to estimate how many safe primes there are below a given number.
jnm2
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Where can I find a list of prime numbers with 17-20 digits?

I see many (non cryptographic) hashes use prime numbers. Where can I find a list of primes that have 17-20digits? An unsigned 64 has 20digits (with the top 2 being 18) so I was thinking 19 digits would all fit and many 20 digits would be usable
Eric Stotch
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Given a prime number x, is there a guarranteed max gap before next Prime Number appears.

Suppose x is an arbitrarily large prime number. Is there a well defined maximum gap before the next prime number y shows up. I am looking for maximum limit of (y - x) in terms of x. Not in terms of n where x may be nth prime number. So, the answer I…
kpv
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A Quest of Legendre Theorem

I had came across a theorem a few weeks ago and I tried to make sense of it but could not at the time, so I put it down and recently picked it back up to take another shot, but the thing is, what I think this problem might involve I am not very…
night owl
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Why are some numbers disproportionately represented when calculating a digit-sum?

I wrote a small python script to calculate a number's digit-sum (i.e #152 = 1 + 5 + 2 = 8) after being raised to various powers. Then I noticed certain numbers are dramatically more common than others, while others don't occur at all. For example,…
A. Salas
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What is the mathematical formula to find the sum of the first 1000 prime numbers?

I am trying to improve my coding skills at codeeval (doing practice problems). One of the programming questions I have to answer is to write code that will sum the first 1000 prime numbers. What is the mathematical formula to find the sum of the…
Jota Onasys
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Why are multiples of 30 adjacent to prime numbers more often than any other number?

This is closely related to this question, in which I asked about pairing of numbers in the same problem, which I defined as follows: Given a number $n$, the sequence of "prime-adjacent multiples" is defined as any multiple $k$ that is $\pm 1$ away…
Recessive
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Can a number have a prime factor that isn't a part of the number's prime factorization?

Is it possible to have a number, $x$, divisible by some prime, such that that prime does not appear in the unique prime factorization of $x$?
oadams
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Totient-like function

I have number written as factors for instance: n = 2 * 3 * 3 * 5. What I have to do is find how many numbers between <1, n) are co-prime to n, which means GCD = 1. It can simply be done using Euler's Totient. But what if GCD = 2 or more? Is there…
Kostek
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Does the sum of the first n primes divide their product more often than not?

The sum of the first n primes often divides their product, but it often doesn't. Which of the two happens most often? Any references to this question?
Bernardo Recamán Santos
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Anomaly of frequency of last digits $1,3,7,9$ in consecutive primes

Consulting https://oeis.org/A032592 shows that there are $35$ quartets of consecutive primes ending in $1,3,7,9$, yet the probability of this is only $1$ in $256$ up to $23869$ which is prime number $2654$. There should be just ten of them from…
J. M. Bergot
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How to find prime numbers

I am looking for a formula that tells me what the next prime number will be. It is hard to do this without a formula because for example there is a small gap between 17 and 19 then a big one between 23 and 29 so it is sometimes hard to find the next…
please delete me
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Special types of prime numbers

We know that there is infinitely many of prime by the proof of Euclidean.and we know also that there are infinitely many of primes in arithmetic progression by the Dirichlet theorem My question is, are there some special types of prime number that…
Abdo
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