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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.

12562 questions
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Next primes of the form $n^n - (n-1)^{n-1} - (n-2)^{n-2} - (n-3)^{n-3} - ...- 3^3 - 2^2 - 1^1 - 0^0$

With the convention $0^0=1$, let $Q(1) = 1^1 - 0^0=1-1=0$ $Q(2) = 2^2 - 1^1 - 0^0=4-1-1=2$ $Q(3) = 3^3 - 2^2 - 1^1 - 0^0=27-4-1-1=21$ $Q(4) = 4^4 - 3^3 - 2^2 - 1^1 - 0^0=256-27-4-1-1=223$ and so on... I found that $Q(2)=2$, $Q(4)=223$ , and…
Dimash K
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Are there always prime numbers that verify these parameters?

Let $p_1$, $p_2$, $p_3$, $p_4$ be prime numbers such that : $p_k\ne 2$ or $3$ for $k=1$ or $4$ $p_1\gt p_3$, $ p_4\gt p_2$ $p_1 - p_3 +2=- p_2+p_4$ Prove that for any $p_1$ and $p_2$, they exist a $p_3$ and a $p_4$. I hope you'll understand ! It…
Jotadiolyne Dicci
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Does there exist a k such that the kth prime is balanced in order k-1?

A balanced prime of order n is a prime number that is equal to the arithmetic mean of the nearest n primes above and below. For example, 5 is a balanced prime in order 1 because it is the average of the prime before it (3) and the prime after it…
histocrat
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Primality of adding digits for 5794651471018341717451336997

On my spare time I was playing with prime numbers and I ended up with this prime: Update : I wrote a small app so far it goes up…
emperon
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Prime Number Solutions

Let p, q, and r be prime numbers such that $$2pqr + p + q + r = 2020$$ Find $$ pq +qr+rp $$ I believe I got the correct answer of $p,q,r$ being $2, 17, 29$ which results in an answer of $585$. I solved it because I knew one of the prime numbers had…
Il Leone
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How to test if a number is not of the form $ p^{2^n} $ where $p$ is a prime number?

I am writing a program to generate a sequence of numbers such that they are not of the form $p^{2^n}$, where $n$ can be a whole number ( i.e $ n \in \{ 0, 1, 2, 3, \ldots \} $ ). I could use the following approach (solve for n): $$ p^{2^n} = N…
deostroll
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Can Three consecutive numbers can be pairwise prime considering middle element is even.

Suppose $a, b, c$ are 3 consecutive integers, and $b$ is an even number. Is it true that the three numbers are pairwise prime? I have tested a few cases, for example $9, 10, 11$, and it seems to be true. How can I prove/disprove that? Thank you.
Deepanshu Singh
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Numbers Divisible by All Primes Less or Equal to Itself

Just to preface, I haven't put too much thought into the following lemma/conjecture: The only number divisible by all primes less or equal to itself is two. Unless I am overlooking something, this conjecture is not as easy to prove rigorously as one…
Venkataram Sivaram
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Is $2^n$ always a sum of exactly $n$ primes for $n \geq 2$?

For example: $2^2=4=2+2$ $2^3=8=3+3+2$ $2^4=16=5+5+3+3$ $2^5=32=17+7+3+3+2$ $2^6=64=17+17+17+7+3+3$ How much further can this go?
user757601
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Find primes satisfying this

Find all prime numbers n such that $$n | 6^{n}(n-4)! + 10^{3n}$$ Can't seem to figure out how to start this, any hint would be helpful. Will Fermat's or Wilson's theorem be used?
user737542
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Prime-quadruples of the form $((2k)!-2k-1,(2k)!-1,(2k)!+1,(2k)!+2k+1)$

I am trying to determine elementarily can there be an infinite number of composite numbers that have at least the two same numbers as two minimal gaps with respect to other composites both from left and from right. I started with the construction of…
user716491
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Show that $p_{k}≥2k-1$ for all $k≥2$

Let $p_{k}$ be the $k^{th}$ prime. Show that $p_{k}≥2k-1$ for all $k≥2$. This inequality is true for several values of $k$. For example $p_{2}=3≥2(2)-1=3$ and $p_{3}=5≥2(3)-1=5$.
Safwane
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Is there a Lay-Mathematician explanation for the proof technique of Zhang's Theorem?

I've heard there are elements of Sieve Theory and whatnot but no further 'outline' of how he proved the theorem.
Joshua Farrell
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Simple formula with high prime output

In doing unrelated calculations I was a bit flabbergasted when I found that the formula $f(n)=8 + 3(5^n)$ gives prime numbers for $n=0,1,2,3$; so I checked further and found: Out of the first 401 natural numbers (that is, $n=0,1,...,400$) 19 of…
MathTrain
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Are there infinitely many primes of the form $9m^2+3m+1$ where $m\in \mathbb{Z}$?

This question is related to Theorem $1$ in my old answer. All primes $p=9m^2+3m+1$ with $m\in \mathbb{Z}$ up to $1000$ can be found in this paper (see Table $1$ on the page $7$).
davidoff303
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