If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships betwe...

An odd prime p is called a Wieferich prime if 2p−1 ≡ 1 (mod p); alternatively, a Wilson prime if (p− 1)! ≡ −1 (mod p). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5, 13, and 563. We report that there exist no new Wieferich primes p < 4×1012 , and no new Wilson primes p < 5×108. It is elementary that both defining congruences above h...

Let ψm be the smallest strong pseudoprime to the first m prime bases. This value is known for 1 ≤ m ≤ 11. We extend this by finding ψ12 and ψ13. We also present an algorithm to find all integers n ≤ B that are strong pseudoprimes to the first m prime bases; with a reasonable heuristic assumption we can show that it takes at most B2/3+o(1) time.

in this paper we will prove that the simple group g2(q) where 2 < q = 1(mod3)is recognizable by the set of its order components, also other word we prove that if g is anite group with oc(g) = oc(g2(q)), then g is isomorphic to g2(q).

in the tradition of philosophy prime matter has been often accepted as a genuine aristotelian concept. however, this has been doubted by some modern scholars. the dispute is mainly whether aristotle committed to prime matter or not. on the other hand, some scholars still endorse prime matter as a part of aristotle’s philosophy. prime matter is tied up to the problem of generation. generation, i...

Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set ofcomplete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets:begin{eqnarray*}V(Gamma) &=& {(x,y) | x in V(X) & y inV(X_x) },\ E(Gamma) &=& {(x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}.end{eqnarray*}The $X-$join graph $Gamma$ is said to be re...

the rings considered in this article are commutative with identity $1neq 0$. by a proper ideal of a ring $r$,  we mean an ideal $i$ of $r$ such that $ineq r$.  we say that a proper ideal $i$ of a ring $r$ is a  maximal non-prime ideal if $i$ is not a prime ideal of $r$ but any proper ideal $a$ of $r$ with $ isubseteq a$ and $ineq a$ is a prime ideal. that is, among all the proper ideals of $r$,...

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...