# 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.

1,833 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
142 views

453 views

### Primes $p$ for which $2p \pm 1$ are also primes

Out of curiosity and trouble sleeping, I decided to look at the distribution of primes $p$ for which $2p \pm 1$ are also primes. I looked at the first 25,910,000 primes and counted the number of ...
277 views

### Relatives of Heegner numbers?

It is well known that Euler's lucky numbers are related to the Heegner numbers, where \begin{align} &n^2+n+p\\ \end{align} gives primes for $n=0,\dots,p-2$ if and only if its discriminant $1-4p$...
300 views

### The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
248 views

### Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
180 views

### Pairs of palindromic primes without $1$ and have a palindromic product

While discussing about prime numbers with other users, I noticed that: $(1)$ There are very few pairs of palindromic prime numbers that do not contain the digit $1$ and that have products which are ...
Let $p$ be a prime number, let $x$ be a variable, and consider two power series over the ring $\mathbb{Z}_p$ of $p$-adic integers: $a(x):=\underset{n\geq 1}{\sum}{\frac{p^n}{n!}x^n}=px+\frac{p^2}{2}x^... 0answers 133 views ### Is there a simple formula for$\binom{2n}{n} \pmod{n^3}$? Is there a simple formula for the following? $$f(n) = \binom{2n}{n} \pmod{n^3}$$ I know$f(n) = 2$iff$n$is prime and greater than$3$, but I don't know anything about composite numbers. 0answers 146 views ### What is the relationship between simple prime-power counting function and$\log\zeta(s)$? This question assumes the following definitions of prime-counting functions where$p$denotes a prime number,$k$denotes a positive integer, and$\theta(y)$is the Heaviside step function which takes ... 0answers 524 views ### Is there a number of the form$f(n)=7k+6=5p$with prime$p$? Here : Numbers$n$of the form$10^{m}(2^{k}−1)+2^{k-1}−1$, where$m$is the number of decimal digits of$ 2^{k-1}$. the numbers $$f(n):=(2^n-1)\cdot 10^d+2^{n-1}-1$$ are introduced , where$d$... 2answers 309 views ### Why the infinity of prime numbers can be proved topologically? I was reading Proofs from THE BOOK by Martin Aigner, Günter M. Ziegler and was very impressed by the following proof of infinity of prime numbers with topology: Edit: The proof can also be found here.... 1answer 188 views ### On prime numbers of the form$7\times10^n+69$and the lights out puzzle Consider those natural numbers$n$such that$7\times10^n+69$is a prime number. The first$15$such numbers are$1$,$2$,$3$,$6$,$7$,$8$,$10$,$12$,$13$,$21$,$46$,$68$,$91$,$153$, and$366$... 0answers 335 views ### Yet an other conjecture about odd numbers$n=a+b$such that$a^2+b^2$is prime This question is related to A conjecture about an unlimited path and Any odd number is of form$a+b$where$a^2+b^2$is prime but I present it on its own if anyone would like to help finding ... 1answer 343 views ### Asymptotic Distribution of Prime Gaps in Residue Classes Define$\pi_{n,a}(x)$as the number of primes$p$less than$x$such that$p\equiv a\bmod n$for coprime$n,a$. This function can be asymptotically approximated by $$\pi_{n,a}(x)=\frac{\operatorname{... 0answers 113 views ### If prime p=a_n10^n+a_{n-1}10^{n-1}+\ldots+a_110+a_0 then f(x)=a_nx^n+\ldots+a_0 is irreducible in \mathbb{Z}[x] I have been trying to solve this problem on my own for four days now, and I cannot figure out how to prove it: If we express a prime p in base 10 as$$p= a_m10^m+a_{m-1}10^{m-1}+\ldots +a_110+a_0,... 0answers 147 views ### Can we prove this conjecture concerning Pell equations? For every positive integer$n$, not being a perfect square, denote the fundamental solution of the Pell equation $$x^2-ny^2=1$$ with$(a,b)$. In other words,$b$is the smallest positive integer such ... 0answers 272 views ### What's infinte sum of the reciprocal of the primorial? $$\sum_{n=1}^\infty \frac{1}{p_n\#} = \frac{1}{2}+\frac{1}{2\times3}+\frac{1}{2\times3\times5}+\dots$$ where$p_n\#$is the nth Primorial. Does this sum approaches some known value or constant and ... 0answers 78 views ### Prove there are infinitely many integer solutions to$z^z = y^y x^x$for with$x,y,z > 1$I have tried a number of methods using prime factorisations but they inevitably lead to invoking too many unknowns for me and balloon in complexity. 0answers 142 views ### Is$\sum_{n=1}^\infty \ln \left(1-\frac{1}{p_n^s} \right)=-\ln \zeta(s)$the only known series with primes and nontrivial closed form? To clarify, I'm asking about series where$n$th term depends only on$n$th prime number. From the famous result (by Euler, I think) we have:$$\sum_{n=1}^\infty \ln \left(1-\frac{1}{p_n^s} \right)=-\... 0answers 129 views ### What is known about the counting function of Gaussian primes" The counting function of primes among$\Bbb{N}$, describing the asymptotic density of the primes, is well known (the Prime Number theorem). Let's define a mild generalization of the counting function ... 0answers 121 views ### Number Theory:$\binom{p-1}{k} \equiv (-1)^k \pmod{p}$I have this problem assigned for homework: Prove that if$p$is an odd prime and$k$is an integer satisfying$1<k<p-1$, then$\binom{p-1}{k} \equiv (-1)^k \pmod{p}$. I've come up with a proof ... 0answers 245 views ### Where is The third Gisella prime? A Gisella prime is a prime number obtained from concatenating the first$n$natural numbers starting from$2$and then replace each composite on that concatenation with the concatenation of its prime ... 0answers 110 views ### Applying iterated function on the sum of the squares of the prime factors of$30$Let$f(n)$denote the sum of the squares of the prime factors of$n$with multiplicity. For example,$f(60)=f(2\cdot2\cdot3\cdot5)=2^2+2^2+3^2+5^2=42$. Denote the iterated function$f^k(n)=\...
Obviously, we do not know an explicit and easily manipulable formula for finding every prime - that is, a function $f(n)$ which yields the $n^{th}$ prime. I've seen plenty of formulas that "cheat" in ...