Zhi-Wei Sun

TL;DR: In this article, the authors studied whether sufficiently large integers can be written in the form cp + Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number.

Abstract: Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $ u_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ and $ u_p(\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k})$ are at least $ u_p(n)$, where $ u_p(x)$ denotes the $p$-adic valuation of $x$. Furthermore, if $p>3$ then $$n^{-1}\sum_{k=0}^{n-1}\frac{\bi{2k}k}{m^k}=\frac{\binom{2n-1}{n-1}}{4^{n-1}} (mod p^{ u_p(m-4)})$$ and $$n^{-1}\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k}=\frac{C_{n-1}}{4^{n-1}} (mod p^{ u_p(m-4)}),$$ where $C_k$ denotes the Catalan number $\binom{2k}{k}/(k+1)$. This implies several conjectures of Guo and Zeng [GZ]. We also raise two conjectures, and prove that $n>1$ is a prime if and only if $$\sum_{k=0}^{n-1}multinomial{(n-1)k}{k,...,k}=0 (mod n),$$ where $multinomial{k_1+...+k_{n-1}}{k_1,...,k_{n-1}}$ denotes the multinomial coefficient $(k_1+...+k_{n-1})!/(k_1!... k_{n-1}!)$.

TL;DR: In this paper, the Bernoulli polynomial of degree n has been shown to be a prime for the harmonic numbers H = √ √ n, where n denotes the degree of degree N. With the help of some combinatorial identities, the following congruences were established:

TL;DR: In this article, it was shown that any n ∈ N can be written as x 2 + y 2 + z 2 + w 2 with x, y, z, w ∈ Z such that x + y + z (or x + 2 y, or x + 3 y + 5 z ) is a square (or a cube).