In this paper we give a survey of recent results on sumsets of subsets of a field with polynomial restrictions. Let F be a field and let F× be the multiplicative group F \ {0}. The additive order of the (multiplicative) identity of F is either infinite or a prime, we call it the characteristic of F . Let A and B be finite subsets of the field F . Set A+B = {a+ b: a ∈ A and b ∈ B} and AuB = {a+ ...

The exotic states $X_{0,1}(2900)$ with the quark flavor of $cs\bar{u}\bar{d}$ are recently observed in mass spectrum $D^+K^-$ $B^-\to D^-D^+K^-$ by LHCb collaboration. To explore nature $X_{0,1}(2900)$, except for analyzing their masses and decay widths as usually did literatures, study production mechanism $B$-meson weak decays would provide another important information. amplitude D^- X_{0,1}...

Let Fp be the field of residue classes modulo a prime number p. In this paper we prove that if A,B ⊂ F∗p, then for any fixed ε > 0, |A + A| + |AB| (

Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only solution \((x,y,z)=(2,2,2)\). this connection we call \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional...

In this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. We prove that if $N$ is a weakly prime subtractive ternary subsemimodule of a ternary $R$-semimodule $M$, then either $N$ is a prime ternary subsemimodule or $(N : M)(N : M)N = 0$. If $N$ is a $Q$-...