5 What is the remainder of $1!+2!+…+ (10^{10})! \mod40$? 5 $\phi(n)^{\sigma(n)^{\tau(n)}}=n^2$ find all natural numbers $n$ such that the equality is true 4 Studying continuity of $f(x,y)=\frac{x+y}{x-y}$ 4 How do I prove that $(3+2\sqrt2)^n (3-2\sqrt2)^n = 1$? 4 What does $\sum_{(a_1,a_2,…,a_n)\in\{1,x\}^n} a_1a_2\cdot\cdot\cdot a_n$ mean?

### Reputation (2,012)

 +10 $\log f \sim \log g$ does not imply $f \sim g$, but what can we say? +10 Space dimension of a $2 \times 2$ matrix -2 Number of partitions of $n$ with Durfee square of size $k$ +10 Prove $+$ and $\times$ are well-defined on quotient rings

### Questions (8)

 3 Polynomials: $f_n \to p$ and $p$ has distinct real roots implies $f_n$ eventually has real roots. 3 $\log f \sim \log g$ does not imply $f \sim g$, but what can we say? 3 A bound for a sum over square-free numbers: $\sum_{n \leq X} \frac{\mu(n)^2 \tau_k(n)}{\phi(n)} \ll (\log X)^k$ 2 Prove $2+2\cos(2\pi\theta) \leq 4\exp(-2\|\theta \|^2)$. 2 In Finite Dimensional $\mathbb{C}[[x]]$ Modules, $x$ is Nilpotent

### Tags (77)

 16 measure-theory × 11 9 real-analysis × 10 14 elementary-number-theory × 4 8 algebra-precalculus × 4 11 number-theory × 7 7 integration × 3 11 calculus × 5 7 multivariable-calculus × 2 10 limits × 6 6 analysis × 5

### Bookmarks (4)

 9 Is $[G_p \cap G_q:G_{pq}]$ always finite? v2.0 2 If $f$ is continuous and bounded, there exists $x_n\to\infty$, with $\,f(T+x_n)-f(x_n)\to 0$ 2 Euler Totient Integer Remainder Sum Lemma 2 Find an integrable function that is not integrable for sufficiently close functions