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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 way to check if a number is prime with only a few details about the number?

I need to find a prime with the info, the sum of all its digits, the first and last digit and how much digits the number has. For example, lets check if $13$ is prime. First digit: 1 Last digit: 3 Sum of all digits: 13 -> 1 + 3 = 4 Amount of digits:…
A...................
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$p(n)/\log p(n)={}$prime

A little item for anyone who wants to test their new exaflop machine. Given some $\operatorname{prime}(n)/\log\operatorname{prime}(n)={}$as near as possible to a prime $q,$ for both log base $10$ and base $e,$ what is $\operatorname{prime}(n)$? …
J. M. Bergot
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sum of the primes with even indices and with odd indices

Referring to https://oeis.org/A077131 and https://oeis.org/A077126 one sees that at a(40) the sum of the primes with odd indices is 7440 and for even indices 7257. Has anyone looked at the possibility of a race between these two, similar to that…
J. M. Bergot
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$\sum_{i=1}^{\infty}\frac{1}{p(i)p(i+1)}$, where $p(i)$ is the $i$th prime number

I know that the sum of the reciprocals of the primes diverges, but I was asked of this question (title). The problem stated that it will converge to a value, but I cannot figure out the value.
David Chen
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It is given that $p = 2q+1$, and $p$ is a prime number. Prove that $(q!)^2 +(-1)^q$ is a multiple of $p$.

It is given that $p = 2q+1$, and $p$ is a prime number. Prove that $$(q!)^2 +(-1)^q$$ is a multiple of $p$. Attempt: $q = \frac{p-1}{2}$ \begin{align} \therefore (q!)^2 +(-1)^q &= \frac{1}{4} [(p-1)!]^2 +(-1)^q \\ &=\frac{1}{4}…
NetUser5y62
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New prime number

I’m just reading some articles on the internet and saw an article that says that there’s found a New prime number. Is that true? Do we last week have found a New one? https://www.ad.nl/economie/grootste-priemgetal-ooit-ontdekt~a33478f2/
WinstonCherf
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Factorizing sum of two powers.

Is it possible to factorize, I'm trying to prove it isn't prime. $x^4 + 15^x$ If for what values of x will the above be prime, also any general method of determining if a really large number is prime? Thanks.
fosho
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is ${(a+b)! \over (a+b)(c!)}-1$= prime number if a,b and c are primes? if 2a=b+c and b>a&c

${(a+b)! \over (a+b)(c!)}-1$ firstly $a,b$ and $c$ need to be prime numbers then the sum of $b+c = 2a$ or double a secondly $b>c$ for example if $a=5, b=7, c=3$ then that gives me $6652799$ which is a prime. However I'm asking if you can prove or…
Thomas Holloway
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Prime numbers in binary.

So I am currently writing a computer program which among other things computes huge binary prime numbers. I am testing it on 16 digit numbers. So here is my question. So I generate 100 random odd numbers (aka a 16 digit string of binary numbers…
Riemann-bitcoin.
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Sequence of non-prime naturals

Regarding the question of finding a sequence of non-prime natural numbers, I have consistently found an answer that states that have a 'proper' starting point, say (n+1)!+2; and then all the consecutive n numbers will be non-prime. I am unable to…
jiten
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Prove that $n$ is prime if and only if for every integer $a$ , either $hcf ( a, n ) = 1 $or $n ∣ a$ assuming n ≥ 2

So the question is an 'if and only if' statement meaning that it needs to be proved both ways. I have proved the first direction (if $n$ is prime then either $hcf(a,n)=1$ or $a∣n$). However, I am struggling to prove the opposite direction.
User19023
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Interpreting Dusart's sum of prime approximation

I am trying to plot Dusart's sum of prime approximation [ref 1]: However I'm not getting the correct sum but very large incorrect numbers. Some things come to mind: 1) I'm entering the acsii version into c as…
onepound
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simply proof of prime number theorem

I will present a very short proof of the Prime Number Theorem. My question is, if the following proof is acceptable? Let ф(np) be the Euler ф function (Euler totient function) for any primorial np with (1) $$np=\prod_{p=prime}^{p≤n} p$$ which is…
Dieter Sengschmitt
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How can I find the number of primes between two numbers

I'm not an advanced mathematician so I'm looking for some easy way . If there is no easy way . Then I'm doomed
esquel
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