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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.
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Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?
Define $f(n)$ to be: $$ \sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d} $$ But $\sigma_0(d) = 2^{\omega(d)}$ for any $d \mid n\#$ a primorial, so: $$ f(n) = \prod_{p \text{ prime} \\ p \leq n} ... elementary-number-theory prime-numbers asymptotics analytic-number-theory arithmetic-functions- 19.2k
Primality test for numbers of the form $\frac{a^p-1}{a-1}$?
This question is cross-linked with Mathoverflow and I didn't get an answer for the question. Here is what I observed : Inspired by Lucas-Lehmer primality test, I think I made a primality test for ... number-theory prime-numbers primality-test- 313
A question on the Riemann zeta function
Question: Consider a $L$-shaped path $L_\epsilon:\frac{1}{2}+\epsilon\to \frac{1}{2}+\epsilon+i\ (H+\epsilon)\to \frac{1}{2}+i\ (H+\epsilon)$ where $H>0$ is fixed and $\epsilon>0$ is arbitrarily ... number-theory prime-numbers analytic-number-theory riemann-zeta l-functionsThree-term arithmetic progressions ending with a given prime
A student has asked me whether the following is true: If p is prime and bigger than 37, there exists a three-term arithmetic progression of primes ending in p. For smaller odd primes one can find 3-... number-theory prime-numbers arithmetic-progressions- 89
Tight lower and upper estimate of alternating summation of reciprocals of divisors of primorial $\sum_{d \mid p_n\#} (-1)^{\omega(d)} \dfrac{1}{d}$?
Let $f(n ) = \sum_{d \mid p_n\#}(-1)^{\omega(d)} \dfrac{1}{d}$ where $p_n$ is the $n$th prime number and $p_n\# = p_n p_{n-1} \cdots p_1$. I'm looking for a way to make a simple estimate of (upper / ... elementary-number-theory prime-numbers upper-lower-bounds estimation primorial- 19.2k
Has Euler's claim about the mystery of prime numbers stood the test of time?
On the door of the math office at my school, there is a picture of Euler, and a quote attributed to him: "Mathematicians have tried in vain to this day to discover some order in the sequence of ... prime-numbers soft-question- 1,324
Solving the equation $p_{a}+p_{a}^{p_{n^{2}-a}}=p_{b}+p_{b}^{p_{n^{2}-b}}$
I'm interested in the solutions of the equation in the title, where $p_{k}$ denotes the $k$-th prime number. This question came to my mind finding out that $5+25=3+27$ where the summands are the ... number-theory prime-numbers diophantine-equations finite-fields- 2,486
If $a^2 +3b^2$ is a prime integer, show that $a +bi\sqrt{3}$ is an irreducible element in $\mathbb{Z}[i\sqrt{3}]$.
If $a^2 +3b^2$ is a prime integer, we have to show that $a +bi\sqrt{3}$ is an irreducible element in $\mathbb{Z}[i\sqrt{3}]$. Now $a^2+3b^2=(a+ib\sqrt{3})(a-ib\sqrt{3})$ as $a^2+3b^2$ is prime, so $a^... abstract-algebra prime-numbers irreducible-polynomials- 147
Knowing the consecutive prime numbers 2,3,...n ; it`s possible to find prime numbers > n and < (n+2) performing operations without dividing $. [closed]
Knowing the consecutive prime numbers $\mathbf{ 2,3,...n}$; it`s possible to find prime numbers $\mathbf{> n }$ and $\mathbf{ < (n+2)^2}$. To do this: Every given prime number can be raised to a ... number-theory prime-numbers- 47
Why do we observe different color densities in columns and rows in different diagonal quadrants within the Ulam spiral?
The attached picture is an Ulam spiral that is coloring all the primes based on the column it resides. For (x, y) coordinates: blue pixels are primes who's values of x are: ..., -6, -3, 0, 3, 6, ... ... prime-numbers- 129
Relationship between prime numbers and powers of 2 [closed]
I understand that Mersenne Primes take the form $ 2^n -1 $ , and that the largest known prime number is a Mersenne Prime. My question is whether there is some upper bound for $c$ such that the set of ... number-theory prime-numbers prime-gaps mersenne-numbers- 9
Are there infinitely many primes of primes?
Primes of primes are prime numbers where each digit is also prime. I have written a computer program to generate such primes from the first 2 billion primes! So far, the program has given me $75,249$ ... number-theory prime-numbers- 383
Divisors on $\mathrm{Spec}(\mathbb{Z})$ and $H^0(\mathrm{Spec} (\mathbb{Z}),D)$ [closed]
Let $D =\displaystyle \sum_{p \in \mathcal{P}} n_p (p)$ a Weil divisor on $\mathrm{Spec} (\mathbb{Z})$, with $n_p \in \mathbb{Z}$. I have some questions about the set $H^0(\mathrm{Spec} (\mathbb{Z}),D)... number-theory algebraic-geometry prime-numbers algebraic-curves divisors-algebraic-geometry- 97
Is my proof that C = {x$\in$ N | x > 1} is smallest superset of the set of prime numbers that is closed under the function f(x,y) = xy correct?
Disclaimer: I didn't formally studied math since high school years ago, so I apologize for breaking any conventions (I'd appreciate pointing that out if I did) I have been self-studying math through ... solution-verification proof-writing prime-numbers- 33
T/F: The interval of width $n$ containing the most amount of primes is $[2,n+2]$?
Given $n\in\mathbb{N},$ the interval of width $n$ containing the most amount of primes is $[2,n+2]$ ( rather than $[2+x, n+2+x]$ ). This sounds like it should be true since the primes spread out more ... prime-numbers- 12.5k
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