In this paper, we introduce and study the prime, strongly prime,semiprime and irreducible fuzzy bi-ideals of a semigroup. We characterize thosesemigroups for which each fuzzy bi-ideal is semiprime. We also characterizethose semigroups for which each fuzzy bi-ideal is strongly prime.

A search for a massless dark photon $\gamma^{\prime}$ is conducted using 4.5 $\mathrm{fb}^{-1}$ of $e^+e^-$ collision data collected at center-of-mass energies between 4.600 and 4.699 $\mathrm{GeV}$ with the BESIII detector BEPCII. No significant signal observed, upper limit on branching fraction $\mathcal{B}(\Lambda^{+}_{c}\to p \gamma^{\prime})$ determined to be $8.0\times 10^{-5}$ 90% confid...

Let \( \{F_n\}_{n\geq 0} \) be the sequence of Fibonacci numbers and let \(p\) a prime. For an integer \(c\) we write \(m_{F,p}(c)\) for number distinct representations as \(F_k-p^\ell\) with \(k\ge 2\) \(\ell\ge 0\). We prove that \(m_{F,p}(c)\le 4\).

• n is prime. Then G is cyclic by Theorem 1. • n = pc for some prime p and c > 1. In this case if there is no element of order n, then all the orders must divide pc−1. We get n = pc elements x such that xp c−1 = 1, violating (*). • n = pq for co-prime p and q. In this case let H and F be two subgroups of G defined as follows: H = {a : ap = 1} and F = {b : bq = 1}. Then |H| ≤ p < n and |F | ≤ q ...