Are there groups, where the conjugacy classes have elements according to the divisors of perfect numbers larger than $6$? I already got the first example, so therefore $6$ is excluded: $S_3$. The corresponding GroupProps page on groups of order $28$ is still empty, but OEIS/A000001 says that there are $4$ groups having $28$ elements. Any […]

Let $n$ be a natural number. Let $\pi_k$ be denoted as follows. $ \pi_{2}(n) = $ the number of twin primes $(p, p+2)$ with $p \le n$. $ \pi_{4}(n) = $ the number of cousins primes $(p, p+4)$ with $p \le n$. $ \pi_{6}(n) = $ the number of sexy primes $(p, p+6)$ with$ p […]

I have a large integer, 422215686281216. I am looking for two fifth powers which when added together, or subtracted from one another, equal this number. How can I tell whether an integer is the sum , or difference, of two fifth powers ?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in $\mathbb{Z}_p$?

A series of questions. Explanations would be useful. I have done the first four parts. I am confused on how to go about the last two. Show that for any prime $p$ the largest power of $p$ that divides $n!$ is $$ \left\lfloor \frac{n}{p}\right\rfloor +\left\lfloor \frac{n}{p^2} \right\rfloor +\cdots +\left\lfloor\frac{n}{p^r}\right\rfloor$$ where $p^r\le n < p^{r+1}$ Use […]

We are given: $n=p^aq^b$ and $\phi(n)$, where $p,q$ are prime numbers. I have to calculate the $a,b,p,q$, possibly using computer for some calculations, but the method is supposed to be symbolically valid and proper. I know that $\phi(n)=p^{a-1}q^{b-1}(p-1)(q-1)$, but I dont know what can I deduct from this.

I’ve found several uses of the phrase “linearly independent over the rationals” that imply that any set of irrationals is linearly independent over the rationals if it is pairwise linearly independent over the rationals, but I can’t find a reference to justify that claim. Can someone point me to such a reference? What I want […]

I need to use the functional equation for $\zeta(s)$ and Stirling’s formula, to show that for $s=\sigma +it$ , with $\sigma <0$: $$ |\zeta(s)| << \left(\frac{t}{2\pi}\right)^{1/2-\sigma}$$ as $t\rightarrow \infty$ (where $\sigma$ is fixed), i.e, $\,\displaystyle{\frac{|\zeta(s)|}{\left(\frac{t}{2\pi}\right)^{1/2-\sigma}}}$ is bounded by some constant that depends on $\sigma$ as $t\rightarrow \infty$. Any reference or the solution itself? I tried […]

If $(a,b) = 1$ then $(a+b,ab) = 1$. If $(a,b) = 1$ then $(a+b,a-b) = 1$ or $2$. If $(a,b) = 1$ then $(a+b,a^2-ab+b^2) = 1$ or $3$. Is there an algorithm to compute the gcd of two polynomials applied to coprime numbers like this?

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t’_n)_{n\geq0}$, there’s a sub-sequence $(t_n)_{n\geq0}$ such that $f(t+t_n)$ converges uniformly in $\mathbb{R}$ to a function $g(t)$ i.e. $$\sup_{t\in \mathbb{R}}|f(t+t_n)-g(t)|\to 0, \ \ when \ \ n\to +\infty.$$ This class of functions was proved to be the […]

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