Narrow Results

Let $1<c<\frac{43}{36}$. In this paper, it is proved that for every sufficiently large real number $N$, the Diophantine inequality

Let I be an ideal of a nearring N. We introduce the notions of equiprime graph of N denoted by EQ_I (N) and c-prime graph of N denoted by C_I (N). We relate EQ_I (N), C_I (N) and the graph of a nearring with respect to an ideal, G_I (N). We prove that diam(EQ_I (N\I)) ≤ 3 and diam(C_I (N\I)) ≤ 3 and deduce that the prime graphs are edge partitionable. It is well-known that the homomorphic image of a prime ideal need not be a prime ideal in general. We study graph homomorphisms and obtain conditions under which the primeness property of an ideal is preserved under nearring homomorphisms.

We show that the set of the numbers that are the sum of a prime and a Fibonacci number has positive lower asymptotic density.

It is proved that if ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_5$ are nonzero real numbers, not all of the same sign and ${\lambda }_1/{\lambda }_2$ is irrational, then for given real numbers $\eta$ and $\sigma$, $0<\sigma <\frac{5}{252}$, the inequality

In this paper, we show that the lower density of integers representable as the sum of a prime and a Fibonacci number is at least 0.0254905.

Let $[𝜃]$ denote the integral part of the real number $𝜃$. In this paper, it is proved that for $2<c<\frac{408}{197}$, the Diophantine equation $[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]=N$ is solvable in prime variables $p_1,\dots ,p_5$ for sufficiently large integer $N$.

In this paper we define prime, semiprime and irreducible ideals in ternary semigroups. We also define semisimple ternary semigroups and prove that a ternary semigroup is semisimple if and only if each of its ideals is semiprime.