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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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Checking divisibility of prime number using digit count

Given a number $n$ and a prime number $p$, we aim to divide $n$ into $r$ digit numbers and sum them to check $n$'s divisibility by $p$. I'm asked to prove that $r$ is a divisor of $p-1$. I think that we need to use fermat's little theorem for this…
Ameen
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Ladder primes, a special class of prime numbers

Introduction In a discussion with a friend we found an interesting problem. I could not identify it in the vast collection of so called super primes. Hence, and for reasons that become clear below, I coined the term ladder primes. If this problem is…
Dr. Wolfgang Hintze
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What values for a prime $p$ are possible if $3p+1$ is a perfect cube?

There is a similar case like this already on the site, but it deals with perfect squares and is relatively easy to solve. But what about perfect cubes? Thus $3p+1= n^3$ ? any help? Thanks!
JCase
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Why there is non finite number of prime number?

How can I prove that there is a non finite number of prime number ? I try to prove it by contradiction but it's not conclusive. Any idea ?
user380364
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Goldbach-like conjecture

While experimenting, I found that every number up to $200000$, with the sole exception of $216$ is the sum of a prime and a triangular number (where $0$ and $1$ are included as prime). Is anything known about this?
J Muscat
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Which of the following is a prime number?

Prime number $a$ can be divided by $3$ with no remainder. Which of the following is a prime number? $$A) a^{2} + a$$ $$B) 2a + 4$$ $$C) a^{2} + 2$$ $$D) a^{3} - 2a^{2} - 2$$ $$E) 4a + 3$$ I assumed $a$ to be $3$ since if it can be divided by $3$ but…
SarpSTA
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prove $p|\left(2+\sqrt{5}\right)^p-2^{p+1}$

Can anyone help me? I know that first part is never an integer and second is, so how is possible that the number which is not an integer is divisible by an integer? p is prime number and p>2
lightning777
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Prove the number of divisors

Can anyone help me to prove this? This is given as a fact, but I don't understand why it is true. For an integer $n$ greater than 1, let the prime factorization of $n$ be $$n=p_1^ap_2^bp_3^cp_4^d...p_k^m$$ Where a, b, c, d, ... and m are…
learning
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Proof of primes of the form $6n+1$

According to OEIS Sequence A002476 (https://oeis.org/A002476), it says that all primes of the form $6n+1$ can be written in the form: $x^2 - xy + 7y^2$ with $x$ and $y$ non-negative. I was wondering if anyone had a proof of this. Thanks!
Anthony Luo
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How far to nearest/next prime?

Is there is metric to know how far we are from the nearest prime number. For example if my number is 38, then we are 3 numbers away from 41? If such a metric doesn't exist, is there an upper bound saying that we must find a prime number before (say)…
TheChetan
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Consecutive prime numbers multiplication pattern

Playing with primes in excel I came to a pattern that I do not understand and I would like to know more about it. Example: |Prime numbers | Multiplies | Subtraction of | Difference of | | | multiplies |…
Radouxca
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Proof that $2^p-2$ is divisible by $p$?

I'd like to prove that $2^p-2$ is divisible by $p$ when $p$ is prime. This is true for thousands of primes $p$ that I tested, but I can't figure out or find a proof.
Jerry Guern
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Determining if $973$ is prime

Without a calculator, determine if $973$ is prime or not I was given this question to solve. I know $973$ is not prime. I was told a strategy to solve whether a number is prime or not is to test all the numbers less than the square root of…
mika
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What is currently the biggest prime number with no smaller undiscovered prime number?

Just out of curiosity, what is currently the biggest discovered prime number with no smaller undiscovered prime number?
muichkine
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Prime numbers, and their digital roots.

Edit It is clear that this conjecture is false, in many, many circumstances, and I am grateful to the whole Math Stack Exchange community for helping me to see this. Thank you! Let $p \in \mathbb{P},$ where $\mathbb{P}$ is the set of prime numbers,…
Taylor
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