for a fixed positive integer , we say a ring with identity is n-generalized right principally quasi-baer, if for any principal right ideal of , the right annihilator of is generated by an idempotent. this class of rings includes the right principally quasi-baer rings and hence all prime rings. a certain n-generalized principally quasi-baer subring of the matrix ring are studied, and connections...

Let p be a prime number, and s a positive integer. Let k be an integer, and let n be an integer such that n ≥ k ≥ p s − p s−1 .

Let n denote a positive integer relatively prime to 10. Let the period of 1/n be a · b with b > 1. Break the repeating block of a · b digits up into b sub blocks, each of length a, and let B(n, a, b) denote the sum of these b blocks. In 1836, E. Midy proved that if p is a prime greater than 5, and the period of 1/p is 2 · a, then B(p, a, 2) = 10 − 1. In 2004, B. Ginsberg [2] showed that if p is...

AbstractThe notion of the prime graph a ring R was first introduced by Bhavanari, Kuncham, and Dasari in 2010. The R, denoted PG(R) is whose vertices are all elements ring, where two distinct x y adjacent if only xRy = 0 or yRx 0. In this paper, we study forms properties integer modulo group, some examples number its spanning trees. it found that for n, maximum degree PG(Z_n) n-1 minimum 1. The...

This paper describes the application of the principles of genetic programming to the field of prime factorization. Any prime factorization algorithm is given one integer and must generate a complete list of primes such that, when multiplied together in varying degrees, produces the original integer. Constructing even a limited factoring algorithm in GP turns out to be extremely challenging and ...

(a) Prove that for each integer n > 1 there are infinitely many (Z/nZ)-extensions of Q ramified at only one prime. (b) Prove that for each integer n > 2 there are no (Z/nZ)2-extensions of Q ramified at only one prime. Why does this not contradict the fact that (Z/pZ)2-extensions of Qp exists for every p? (c) Given an explicit example of a (Z/2Z)2-extension of Q ramified at only one prime (with ...

In this paper we study mixed sums of primes and linear recurrences. We show that if m ≡ 2 (mod 4) and m+1 is a prime then (m2n−1−1)/(m−1) 6= m+p for any n = 3, 4, . . . and prime power p. We also prove that if a > 1 is an integer, u0 = 0, u1 = 1 and ui+1 = aui + ui−1 for i = 1, 2, 3, . . ., then all the sums um + aun (m, n = 1, 2, 3, . . .) are distinct. One of our conjectures states that any i...

Given an integer n and a group G we estimate the probability that a random, monic, degree n, p-adic polynomial has Galois group isomorphic to G. A comparison is made with classic results concerning Galois groups of random, monic, degree n polynomials with integer coefficients. We conclude by analyzing the probability that a random, monic, degree n polynomial in Z[x] has these properties: its Ga...

We were well programmed long ago to do this arithmetic, and yet it is quite the most mysterious subject. A positive integer is said to be a prime if it is not equal to one, and if it has no integer factors other than itself and one. Here we see that 12 is factored into the product of the prime numbers 2 and 3. Factorization into primes is unique (a nice matter to prove and one of the first basi...