Articles of number theory

Are there groups with conjugacy classes as large as the divisors of perfect numbers?

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 […]

Asymptotic expressions of $\pi_{2}(n), \pi_{4}(n)$ and $\pi_{6}(n)$

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 […]

Can 422215686281216 be expressed as the sum or difference of two fifth powers?

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 ?

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

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 $\mathbb{Z}_p$?

Number of primes less than 2n

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 […]

Factoring a number $p^a q^b$ knowing its totient

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.

Reference for linear independence of a set of pairwise-independent irrationals

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 […]

Some asymptotics for zeta function.

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 […]

An Algorithm to compute the GCD of polynomials of coprime numbers?

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?

Non uniform continuity of a function and almost periodicity

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 […]