Finiteness of p-Divisible Sets of Multiple Harmonic Sums

On 2-adic orders of some binomial sums

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$ \sum_{k=-n}^n\binom{2n}{n-k}k^{2r} $$ is divisible by $2^{2n-\min\{α(n),α(r)\}}$, where $α(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.

通用的几乎质数长度的广义多边形数之和

Universal sums of generalized polygonal numbers of almost prime length

本文研究了广义多边形数的通用和，证明了在输入素因子数量受限的情况下，通用和的有限性定理，并给出了不同条件下的最优界。

递推序列项之和与范数型方程解集中的$S$-单位

Sum of terms of recurrence sequences and $S$-units in the solution sets of norm form equations

本文研究范数型方程解的结构，证明线性递推序列项之和及固定项数$S$-单位之和表示的解均具有有限性。

On Linnik's conjecture: sums of squares and microsquares

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a conjecture of Linnik for almost all natural numbers, and sharpens a conclusion of Bourgain, Rudnick and Sarnak concerning nearest neighbour distances between normalised integral points on the sphere.

Two Restricted ABC Conjectures

Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields.

算术级数的（交错）幂和多项式值的有效结果

Effective results for polynomial values of (alternating) power sums of arithmetic progressions

本文针对满足特定互质与阶数条件的算术级数，证明了其常规幂和与交错幂和多项式值的有效有限性结果。