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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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Ivy/APL primes algorithm

The Google’s Ivy interpreter is a APL-like math language. This is an example of first N primes numbers algorithm in Ivy/APL op primes N = (not T in T o.* T) sel T = 1 drop iota N Could someone describe the notation for the primes algorithm…
loretoparisi
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Prime numbers ending digits, which is the most abundant?

Prime numbers can end in 1,3,5, 7, 9. I wonder, which ending digit has the most abundant prime numbers, is there any proof?
an offer can't refuse
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Can we deduce that $\lim_{n\to\infty}(\log(p_{n+1})/\log(p_{n}))=1$

Let $p_{k}$ denotes the sequence of prime numbers. We know that $$\lim_{n\to\infty}(p_{n+1}/p_{n})=1$$ Can we deduce that $$\lim_{n\to\infty}\log(p_{n+1})/\log(p_{n})=1$$ where $\log$ is the natural logarithm.
Safwane
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What is the probability that a randomly chosen number N satisfying N ≡ 1 (mod 3) is prime?

What is the probability that a randomly chosen number N satisfying $N ≡ 1$ mod 3 is prime? I know that the "probability" that a randomly chosen m $\in \mathbb{Z}$ is prime is $\frac{1}{ln(m)}$ and that $N = 1 + 3k$ for some $k \in \mathbb{Z}$ But…
Doldrums
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The numbers a and b are simple numbers between them.

The numbers a and b are simple numbers between them. They have no common divisors other than number 1. We know that a/b=1,(3). Find a^2-b^2. So I tried a=1,(3)*b and then substituted into the expression. [1,(3)b-b][1,(3)b+b] But it didn't lead me…
Maybeline Lee
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Characteristics of a Prime Number

Will it be true if I say "If a number is not divisible by any of the numbers from $2$ to $9$, it is a prime number." If no, can you mention some numbers which defy this statement?
Bob
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How to find the smallest possible $n\in\mathbb{N}$ so that $n=2k_1+1=3k_2+2=4k_3+3$?

I start solving this by trying $2k_1+1=3k_2+2$, but it doesn't make any sense, because I still have to use $4k_3+3$. If I sum all three equations together, I get $3n=(2k_1+3k_2+4k_3)+6$. This may be, but how can I determine the factors $k_1, k_2…
thunder
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Test for primality by using numbers 2 to 9

What is a number that is not prime and is not divisible by numbers 2 to 9? If a number is not divisible by numbers 2 to 9 can i say it's a prime number?
Zetaku
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infinite chains of primes?

so my question is if this has been considered imagine a big prime $ q $ then my uqestion is if we can find an infinite group of primes so $ P_{n}=P_{n+1}+k $ with $ P_{0}=q $ and k a positive integer this would mean that we can find an infinite…
Jose Garcia
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Find all base $b$ of both pseudoprimes $15$ and $21$

Find all bases $b$ such that $15$ and $21$ are pseudoprimes,i.e. $b^{14} \equiv 1 (mod 14)$ and $b^{20} \equiv 1 (mod 20)$ Can anyone help me?
Muniain
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Find all the numbers N,P that satisfy the following conditions

Find all the numbers $N$, $P$ that satisfy: $N$ is a whole number $P$ is a prime number $N = \frac{2P^2 -2P}{P+1}$
Daniw
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Prove that there is infinitely many indices $n$ such that $p_{n+1}$ is not of the form $p_{n}+2d$

Let $(p_{n})_{n≥1}$ be the sequence of prime numbers. Then my question is: How one can proves that there is infinitely many indices $n$ such that $p_{n+1}$ is not of the form $p_{n}+2d$. Already we know a finite set of those indices. Here $d≥1$.…
Safwane
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How do I solve $3xy=4xz=5yz$ ${x+y \over x-z}=?$

$3xy=4xz=5yz$ $ {x+y \over x-z}=?$ The answer is ${9 \over 2}$ But how to solve? I am preparing for exam. This question comes from metropol mathematics 1 testbook
black
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What are we looking for when we are trying to prove Riemann Hypothesis?

My knowledge of mathematics is fairly limited. But I know that Sieve of Erastothenes can be used to find prime numbers in increasing order. While Riemann Hypothesis enforces that distribution of prime numbers is not random. But I still don't get it,…
Rajat Pundir
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What is the name of rank $n$ where these primes occurs

Let $(p_{n})$ be the sequences of primes. We know that the quantity $g_{n}=p_{n+1}-p_{n}$ is called gap. Consider the case of twin primes $p_{n+1}-p_{n}=2$. My question is: What is the name of rank $n$ where these primes occurs. For example,…
Safwane
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