# Tagged Questions

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.

442 views

### Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $q\equiv m!\pmod p$ for some $2<m<p$?

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
225 views

130 views

### Discrete log for prime powers

I was fiddling around and found that a function of the form $$L_b (x)=\left(\frac{b^{\phi (p^k)}-1}{p^k}\right)^{-1}\left(\frac{x^{\phi (p^k)}-1}{p^k}\right) \mod p^k$$ seems to behave similarly to a ...
111 views

### The greatest common divisor of multiple numbers

What is the cardinality of the following set $\{{\bf x}=(x_1,\ldots,x_d): \text{each } x_i\in \{ 1,\dots,n \},\text{ and } \gcd({\bf x})=1\}$, where $\gcd({\bf x})$ is the greatest common divisor of ...
30 views

### For which $0\leq a<p^2$, where $p$ is an odd prime, we have that $(2p-1)!\equiv a\mod{p^2}$

Let $p$ be an odd prime. I need to find for which $0\leq a < p^2$, $(2p-1)!\equiv a\mod{p^2}$. If $a\equiv (2p-1)!\mod{p^2}$, then we have that $a = kp^2 + (2p-1)!$, and therefore $p\mid a$, ...
54 views

### Prime Number Theorem and the Riemann Zeta Function

Let $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function. The fact that we can analytically extend this to all of $\mathbb{C}$ and can find a zero free region to the left of the ...
41 views

### Show that for $x,y,z\in\mathbb{Z}$, if $x$ and $y$ are coprime, then $\exists n\in\mathbb{Z}$ such that $z$ and $y+xn$ are coprime.

Not sure where to start on this one. I understand that coprime indicates that their GCD is 1, but I am somewhat confused how to proceed.
74 views

### Is one of $k+1^2,$ $k+2^2,$ …, $k+N^2$ always prime?

I know that the Bunyakovsky conjecture is still open, so we can't prove that there exist primes of the form $n^2+k$ for a given $k$. But suppose that they do: is the least $n$ such that $n^2+k$ is ...
62 views

45 views

### Heegner Prime visualizations

The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ring of integers $\textbf{Q}(\sqrt{-d})$ have unique factorizations. 1 gives the Gaussian integers. 3 gives the Eisenstein integers. 7 ...
1k views

### Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ (Fermat'...
54 views

### No primes in $f(n) = n(n+3)/2$ except 2 and 5

How can I prove that the sequence $f(n) = n(n+3)/2 = 0, 2, 5, 9, 14, 20, 27, 35, 44, ...$ does not contain primes except $2$ and $5$?
52 views

### Is this possible to solve? [closed]

If a, b, c are different prime numbers such that (a-b)(a-c) = 255, find the value of b + c.
210 views

### Percentage of Composite Odd Numbers Divisible by 3

What is the percentage of odd composite positive numbers divisible by 3? In that same vein, what is the percentage of odd composite positive numbers divisible by 5? And, for the future, what is the ...
64 views

### Does the PNT establish a connection between primes and the logarithm?

The prime number theorem states that $$\pi(x) \sim \frac x {\ln(x)}$$ Morally, this seems to suggest that there is a fundamental connection between primes and the natural logarithm. But since we're ...
### How to count the number of perfect square greater than $N$ and less than $N^2$ that are relatively prime to $N$?
I know a little about Euler's totient function that counts integer less than $N$ that are relatively prime to $N$. But I don't know how to modify the function for perfect square numbers, or maybe ...
### For any $x\in \mathbb{N}$ does there exist $m\in \mathbb{N}$ such that $2x+1+2m, 2x+1+4m$ are both prime?
Could someone please give me a proof (or counter example) for this (I believe it is true): For any $x$ (Whole Number) there exists some $m$ (Also Whole) such that $2x+1+2m$ and $2x+1+4m$ are both ...