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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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How to split a list in n parts so the calculation time will be equal

I'm trying to implement a prime number finder. It as to find primes from 0 to X. I use this algorithm (performance may be questionable but this is not the question) to find the primes : prems.Add(3); // Filling the list for (int i = 5; i <…
Thomas Ayoub
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What is the number of digits of this number: $2^{333111160}$?

My question is: What is the number of digits of this number? : $$2^{333111160}$$
DER
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About a recurrence equation of prime numbers

Let $p$ be a prime. Consider the recurrence equation $$s_{n}=(s_{n-1}²-2)(mod(2^{p}-1))$$ where $s₀=4$ My question is: Can we write this recurrence as follow? $$s_{n}=(2^{p}-1)q+(s_{n-1}²-2)$$ where $q$ is an integer.
DER
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Primes Involved in GCD

If p is a prime number, prove that gcd(p, (p-1)!) = 1 So, I've tried using the fact that 1 = px + (p-1)!y, where x,y are integers, but from there I'm stuck and don't really know how to work with the factorial term.
Ian Murphy
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How do we identify the $n$th prime?

The 1st prime is 2. The 2nd prime is 3. The 3rd prime is 5. So if $\pi(x)=1$, the prime is 2. If $\pi(x)=2$, the primes are 2 and 3, but how do we identify them? If $\pi(x)=3$, I know that there are 3 primes, but how can I find what each one is? Of…
Jeffrey Young
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Can two different sets of prime numbers sum up to the same value

Assume A is a set of prime numbers with no duplicate elements B is a set of prime numbers with no duplicate elements A is not equal to B Cardinality of A is equal to cardinality of B Is it possible for sum of the elements of A to be equal to sum…
Logan
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If an integer $a$ is coprime with an integer $b$, then will the integer $a$ also be coprime with the integer $9a + b$?

If an integer $a$ is coprime with an integer $b$, then will the integer $a$ also be coprime with the integer $9a + b$ ?
greese
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Probability with Primes

What is the probability that a positive divisor of $8748$ million is the product of exactly $20$ non-distinct primes? To try and solve this question I split up $8748$ into $2^8 \cdot 5^6 \cdot 3^7 $ and came up with $4/(9 \cdot 8 \cdot 7)$ as an…
user161367
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Probability and Prime Numbers

What is the probability that a positive divisor of 8748 million is the product of exactly 20 non-distinct primes?
user161367
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Existence of at least one prime for all sequences in the family of sequences

Prove or disprove that for a fixed $n \in N$, there exists at least one prime among the integers of the form $2^{k}n+2^k-1$ for an arbitrary $k \in N$.
tesgoe
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How to prove this modular criterion for prime numbers of the form $p=2^n \pm a$?

How to prove following statement : For prime numbers $p$ greater than $3$, it is true that: if $p=2^n-a$ and $a\equiv 1 \pmod 6$ then $p\equiv 1\pmod 3$ if $p=2^n+a$ and $a\equiv 5 \pmod 6$ then $p\equiv 1\pmod 3$ if $p=2^n-a$ and $a\equiv 5 \pmod…
Pedja
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Cut a piece of dough into $3$ even pieces

How could you cut a piece of dough into $3$ even pieces? Cutting it into $2$ is easy, but it's not that trivial for greater numbers. If you can cut it into $n$ pieces, you could repeat the process on each piece, getting any multiplication of $n$, so…
maxuel
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A series that has prime numbers as a element

I was looking at the following sequence 3,5,7,9,11,13,15,17,19,21,23,... The terms are given by $ a_n=n^2-(n-1)^2 $. When I expanded the sequence I noticed that it contained all the prime numbers between $ 2 $ and $ 100 $ Will the all the prime…
Gobabis
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Relate $n$ with $2$

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^n+b^n$ is prime, what can you relate $n$ with 2? My approach: for $a^n+b^n$ to be prime $\forall n>1$, $a$ and $b$ has to be coprimes. But how do I ascertain anything about…
Ruddie
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