We continue to explore cyclotomic factors in the descent set polynomial Qn(t), which was introduced by Chebikin, Ehrenborg, Pylyavskyy and Readdy. We obtain large classes of factors of the form Φ2s or Φ4s where s is an odd integer, with many of these being of the form Φ2p where p is a prime. We also show that if Φ2 is a factor of Q2n(t) then it is a double factor. Finally, we give conditions fo...

In this paper, a new factorization algorithm is presented, which finds a prime factor p of an integer n in time (D log n)O(1), if 4p − 1 = Db2 where D and b are integers. Hence this algorithm will factor a number efficiently, if it has a prime factor p such that 4p−1 is a product of a small integer and a square. Such primes should be avoided when we select the RSA secret keys. Some generalizati...

Previously we have described a mutant Hfr strain in which incompatibility between the integrated F factor and an autonomous F-prime (F') factor was abolished. The mutation (inc) was located in the integrated F factor. F-prime factors isolated from the mutant Hfr strain have the same incompatibility behavior as those isolated from normal Hfr strains. Reintegration of these F' factors into the ch...

Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p′ are connected in Γ(G), whenever G has an element of order pp′. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G) = Γ(P ), G has a composition factor isomorphic to P . In [...

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  ...

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

Let P (m) denote the greatest prime factor of m. For integer a > 1, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, P (a − 1) ,a n2− for any > 0. We study analogues results for the corresponding divisibility sequence over the function field Fq(t) and for some divisibility sequences associated to elliptic curves over the rational field Q. In honor of M. Ram Murt...

Given an integer n ≥ 2, let p m (n) denote the middle prime factor of n. We obtain an estimate for the sum of the reciprocals of p m (n) for n ≤ x.

This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived. Moreover, since most graphs are prime although they can have a product-like structure, also known as approximate graph products, the practical application of the...