144 Can the product of infinitely many elements from $\mathbb Q$ be irrational? 78 Compare two powers of numbers without common divisor 34 Passwords: Two 50-characters vs one 100-characters 13 Do there exist any integer solutions for $y=\log_2(1+3^x)$? 12 Can both $n+3\; \text{and}\; n^2+3$ both be cubic numbers at same time?

### Reputation (6,938)

 +10 Prove that $\{(1 2),(1 2 ... n)\}$ can generate $S_n$ +10 How many sets $(A_1,A_2,\cdots, A_k)$ which are subsets of $\{1,2,\cdots ,n\}$ +10 Can the product of infinitely many elements from $\mathbb Q$ be irrational? +10 Let $G$ be a finite matrix group in $GL_2(Q)$ such that every matrix $A\in G$ has integer entries. Prove $A^{12}= I$ for each $A$.

### Questions (6)

 10 Are there infinitely many natural numbers $n$ such that $\mu(n)=\mu(n+1)=\pm 1$? 8 Is $\mathbb{Z}_2 \times \mathbb{Z}_4^\infty$ isomorphic to $\mathbb{Z}_4^\infty$? 6 Functions $f$ such that $f(x+1)-f(x-1)=2f'(x)$. 6 What are the conditions on $a$ such that the polynomial $x^4-2ax^2+x+a^2-a$ has four distinct real roots? 3 In a metric space $(X, d)$, if closed sets $A$, $B$ contain sequences $a_n,b_n$ such that $d(a_n,b_n)\to 0$, must $A\cap B\neq \emptyset$?

### Tags (100)

 160 sequences-and-series × 8 78 arithmetic 144 irrational-numbers 73 functional-equations × 17 94 inequality × 9 72 combinatorics × 13 88 elementary-number-theory × 29 57 contest-math × 15 84 exponentiation × 2 52 number-theory × 17

### Bookmarks (2)

 13 How to prove there exist two elements in a positive integer sequence with bounded differences such that one is a multiple of the other? 8 $af(a)+bf(b)+2ab$ is a perfect square for all $a,b \in \mathbb N$