Is it possible that the lengths of two sides of a primitive Heron triangle have a c ommon factor? The triangle with sides 9, 65, 70 and area 252 shows that this is possible. But notice that the common factor 5 is a prime of the form 4+1. Surprisingly, however, such a common factor cannot be a prime of the form 4 ; 1. Heron's name is familiar to anyone who has used the formula = p s(s ; a)(s ; b...

Denote by c = 2א0 the cardinal of continuum. We construct an intriguing family (Pα : α ∈ c) of prime z-ideals in C0(R) with the following properties: • If f ∈ Pi0 for some i0 ∈ c, then f ∈ Pi for all but finitely many i ∈ c; • T i6=i0 Pi 6⊂ Pi0 for each ı0 ∈ c. We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type κ of prime z-ideals in C0(...

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b in R$ with $ab in P-IP$ implies either $a in P$ or $b in P$. We give some characterizations of $I$-prime ideals and study some of its properties.  Moreover, we give conditions  ...

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...

We characterized prime-boost vaccine regimens using heterologous and homologous vector and gene inserts. Heterologous regimens offer a promising approach that focuses the cell-mediated immune response on the insert and away from vector-dominated responses. Ad35-GRIN/ENV (Ad35-GE) vaccine is comprised of two vectors containing sequences from HIV-1 subtype A gag, rt, int, nef (Ad35-GRIN) and env ...

The mass relation $${M_{0^{+}}+3M_{1^{+\prime }}+5M_{2^{+}}= 9M_{1^{+}}}$$ miraculously holds for the P-wave charmonium $$(c\bar{c})$$ and bottomonium $$(b\bar{b})$$ systems with soaring precision. origin of such can be addressed from Quark Models, have been confirmed experimentally in a limited number cases. In this connection, we propose $$M_{0^{+}}+5M_{2^{+}}=3(M_{1^{+\prime }}+M_{1^{+}})$$ ...

We prove that a graph G contains no induced 5-vertex path and no induced complement of a 5-vertex path if and only if G is obtained from 5-cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement (where split unification is a new class-preserving operation that is introduced in this paper).

Zaremba conjectured that given any integer m > 1 , there exists an integer a < m with a relatively prime to m such that the simple continued fraction [0, cx.cr] for a/m has c¡ < B for i = 1, 2,..., r, where B is a small absolute constant (say 5 = 5). Zaremba was only able to prove an estimate of the form c, < C log m for an absolute constant C . His first proof only applied to the case where m ...

 Prime submodules and artinian prime modules are characterized. Furthermore, some previous results on prime modules and second modules are generalized.

Praeludium We will call prime numbers the integers n ≥ 2 which are divisible only by 1 and themselves. Euclid (fourth century B. C.) first showed that there exist infinitely many prime numbers. His proof is an excellent example of a mathematical argument: if 2, 3, 5, . . . , p were the only prime numbers, we could construct the number 2 · 3 · 5 · · · p + 1, which has the property of leaving the...