MathOverflow

If $f(x,y)\in\mathbb Z[x,y]$ is a bivariate polynomial is there an algorithm or tool to test for its irreducibility efficiently?
Mathematica only divides by $4$ for this polynomial:$$37822862580314550854496242910543819448902818552410000 *x*y + 9455715645078637713624060695571877269453634487094336 *(x^2) + 9455715645078637713624061084533893822816083009284964 *(y^2) + 3782286258...

Let $(F_k)_{k \geq 1}$ be the Fibonacci sequence with $F_1 = 1$, $F_2 = 2$. By Zeckendorf's theorem, every positive integer $n$ has a unique representation$$n = F_{k_1} + F_{k_2} + \cdots + F_{k_s}, \quad k_1 < k_2 < \cdots < k_s, \quad k...

Question: For given $k\in\mathbb{N}$, is it true that there are only finitely many $n\in\mathbb{N}$ such that $k\le \tau(n)$ and$$n=\displaystyle\sum_{i=1}^k{d_i}^2$$where $\tau(n)$ denotes the number of divisors of $n$, and $d_1<d_2<...<...

Alice and Biboo play a game. Each privately rolls a fair $n$ sided die labelled with $\{0, ..., n-1\}$, visible only to themselves. Players take turns with Alice starting first. Alice starts by making an non negative integer bid. Biboo must then either...