On some universal sums of generalized polygonal numbers

For $m=3,4,\ldots$ those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\in\mathbb Z$ are called generalized $m$-gonal numbers. Sun [13] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\mathbb Z$ (i.e., any $n\in\mathbb N=\{0,1,2,\ldots\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\in\mathbb Z$). We prove that $p_5+bp_5+3p_5\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\in\mathbb N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\in\mathbb N$; in contrast, we show that $p_3+p_5+p_{11}$ and $3p_3+p_5+p_7$ are universal over $\mathbb Z$. Our proofs are essentially elementary and hence suitable for general readers.Comment: Final published versio

Reliable Energy-Efficient Routing Algorithm for Vehicle-Assisted Wireless Ad-Hoc Networks

We investigate the design of the optimal routing path in a moving vehicles involved the internet of things (IoT). In our model, jammers exist that may interfere with the information exchange between wireless nodes, leading to worsened quality of service (QoS) in communications. In addition, the transmit power of each battery-equipped node is constrained to save energy. We propose a three-step optimal routing path algorithm for reliable and energy-efficient communications. Moreover, results show that with the assistance of moving vehicles, the total energy consumed can be reduced to a large extend. We also study the impact on the optimal routing path design and energy consumption which is caused by path loss, maximum transmit power constrain, QoS requirement, etc.Comment: 6 pages, 5 figures, rejected by IEEE Globecom 2017,resubmit to IEEE WCNC 201