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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 recursive sequence for prime numbers invented!

I invented a recursive sequence below nearly 10 years ago, which assigns unity to every non-prime number and also gives all prime numbers. Do you think it is useful?! Starting with $a_0=2$, we…
Mohammad Javanshiry
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Ulam Spiral mod p or mod p-twin

I’ve been looking for anyone who has done this and can’t seem to find it. Would love some help with the visualization. Create an Ulam spiral, but instead of using integers, use only primes. I did this by hand out to around 700 and noticed that there…
Jarod Yates
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What does "up to" imply in Riemann's paper?

I was reading Riemann's paper, On the number of primes less than a given quantity, and found the following pharagraph. The known approximating expression $\pi(x) = Li(x)$ is therefore valid up to quantities of the order $x^{1\over2}$ and gives…
비선형편미분방정식
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Equal remainders from four consecutive primes $p_1,p_2,p_3,p_4$ when $p_2-p_1=p_4-p_3$ in $\frac{p_2 \cdot p_3}{p_1}$ and $\frac{p_2 \cdot p_3}{p_4}$

Given four consecutive primes $p_1, p_2, p_3, p_4$ with $p_2-p_1=p_4-p_3$ and the division of $p_2*p_3$ by $p_1$ and $p_4$ giving the same remainder, is there a relationship between the value of these remainders and the pattern of gaps between the…
J. M. Bergot
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$p$ ways to write $p$ as sum of primes

I hope this question is valid as I'm just curious. in a tweet from AlgebraFact I read the following: "There are 17 ways to write 17 as a sum of primes": $17, 2+2+13, 3+3+11, 3+7+7, 5+5+7, 2+2+2+11, 2+3+5+7, 2+5+5+5, 2+2+3+3+7, 2+2+3+5+5, 3+3+3+3+5,…
Luis Alexandher
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How to check if a large number is prime

When I say large numbers I mean large like 512 (or potentially 1024 or 2048) significant bits. These numbers are so large that there are a multitude complications. For example with python, which supports arbitrarily large integers, checking…
kpie
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What do you mean by 3 consecutive odd prime integers?

I have a confusion over understanding of the statement "3 consecutive odd prime integers". I was looking at a question, where they considered 3 consecutive odd integers that are prime (in this case only 3, 5, 7 will satisfy as after this set every 3…
Goutham
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Are there infinitely many primes of the form $\ p_1p_2\ldots p_k+1\ $ where all the $\ p_i\ $ are distinct prime numbers?

A common misinterpretation of Euclid's original proof for the infinitude of prime numbers is that numbers of the form $\ p_1p_2\ldots p_k+1\ $ (where the $\ p_i\ $ are distinct) must be prime. I understand this is not what the proof says. However,…
Adam Rubinson
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Can you prove a number is prime with evidence?

Hypothetically, I am 100% confident that a large multi-million digit number is prime because I expended the resources to run an extensive primality test. Let's say the number Truly is a prime. Is there any proof I could provide that would save…
Corey Ogburn
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Potential pattern in searching for Cunningham chains?

Say we have a function (loosely speaking) which takes an input of an integer x1x2x3x4..xn, where xn is the nth digit in the integer. This function sums the digits of this number modulo(9) and maps it to this number. So f(19)-> (1+9)modulo 9 -> 1 ,…
user2814916
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Prove or disprove: There exists $a \in \mathbb N$ such that for all $n \in \mathbb N$, $an+1$ is prime.

I'm working on disproving this statement but am having an issue figuring out where to start. I considered splitting it into cases, but so far from that I've only been able to prove the parity of the value, not whether it's composite or prime. Any…
mattbea
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Quirky Primes Test

Wrote a program to list all primes $n$ below $1,000,000$ and noticed they never satisfied (except $n=5$) $\frac {n-1}2 \equiv 2 \pmod{10}$ or $\frac {n-1}2 \equiv 7 \pmod{10}$ Is that true for all primes? Thanks. #include #include…
vengy
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Seeking for an improvement of: $ P_n \sim \frac{10^n[\ln(n)-\frac{1}{4}]}{2\operatorname{Li}(10^n)}$

$$P_n \sim n\ln(n)\tag1$$ Where $P_n$ is the nth prime number Equation $(2)$ is an improvement of $(1)$ $(2)$ comes about through trial and error using Wolfram alpha. $$ P_n \sim \frac{10^n[\ln(n)-\frac{1}{4}]}{2\operatorname{Li}(10^n)}\tag2$$ Where…
Sibawayh
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Tight bounds on the prime counting function

What are the best bounds for $\pi(x)$ i.e. the number of primes less than or equal to $x$ ? From Wikipedia I saw that: $$\frac{x}{\ln x}\left(1 + \frac{1}{\ln x}\right) < \pi(x) < \frac{x}{\ln x}\left(1 + \frac{1}{\ln x} + \frac{2.51}{(\ln…
Vor
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Characteristics of a Prime - Prime Number Game

Do prime numbers have defined characteristics like say perfect squares do? Without seeing the complete string of digits it is relatively easy to determine if a number is definitely not a perfect square by just looking at the least significant…
DeveloperChris
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