On Generalized Mersenne and Fermat Primes

The classical Merseniie and Fermat primes are, respectivel3^ primes of the form 2^\u27 - 1 and 2^‘ + 1. The Mersenne primes have been studied since antiquity. It is known that if 2^ - 1 is prime then k is prime. As of September 2008, there are forty-six such primes known. Fermat primes, of the form 2^’ -h 1, seem to be more rare. It is known that if 2^\u27 -I- 1 is prime, then k must be a power of 2. To date only 2“*^ -I- 1, 2^* -f 1, 2^^ -f-1, 2\u27^^ -t-1, and 2^^ -|-1 are known to be prime. My work involves generalized Mersenne and Fermat primes. Definition: If 6^\u27 — is prime, where a,b, and k are positive integers with a \u3c b and A: \u3e 3, then is a generalized Mersenne prime. I have been able to prove the following analogues to known theorems on Mersenne primes. Theorem: If 6^’ - is a generalized Mersenne prime, then i) = a -t- 1 and ii) k is prime. Theorem: If p is prime and q is a prime divisor of (a -h 1)^ - then q = I (mod p). Using Mathematica, I have found tens of thousands of generalized Mersenne primes. Definition: If is prime, where a, b, and k are positive integers and k \u3e 2. then + b^ is a generalized Fermat prime. Concerning these primes, I have proven the following. Theorem: If a 7^ 1 and -h 6^’ is prime, then k is a power of 2. and a ^ b (mod 2). While there are only five known classical Fermat primes. I have found thousands of generalized Fermat primes. 11 Ill considering whether or not a number is prime, the following is helpful. Theorem: If q is a prime factor of + 6^^, then q = I (mod It is of interest to note that for generalized fermat primes the condition b = a+1 is not necessary as is seen with such examples as 1Q2 + 12 = 101 6^ + 1^ = 1297 5^ + 2^^ = 641. I continue to investigate these and other primes of special forms

The Characterization Of Graphs With Small Bicycle Spectrum

Matroids designs are defined to be matroids in which the hyperplanes all have the same size. The dual of a matroid design is a matroid with all circuits of the same size, called a dual matroid design. The connected bicircular dual matroid designs have been characterized previously. In addition, these results have been extended to connected bicircular matroids with circuits of two sizes in the case that the associated graph is a subdivision of a 3-connected graph. In this dissertation, we will use a graph theoretic approach to discuss the characterizations of bicircular matroids with circuits of two and three sizes. We will characterize the associated graph of a bicircular matroid with circuits of two sizes. Moreover, we will provide a characterization of connected bicircular matroids with circuits of three sizes in the case that the associated graph is a subdivision of a 3-connected graph. We will also investigate the circuit spectrum of bicircular matroids whose associated graphs have minimum degree at least i for k ≥ 1, and show that there exists a set of bicycles with consecutive bicycle lengths

Zero Divisor Graphs and Poset Decomposition

A graph is associated to any commutative ring R where the vertices are the non-zero zero divisors of R with two vertices adjacent if x · y = 0. The zero-divisor graph has also been studied for various algebraic stuctures such as semigroups and partially ordered sets. In this paper, we will discuss some known results on zero-divisor graphs of posets as well as the concept of compactness as it relates to zero-divisor graphs. We will dicuss equivalence class graphs defined on the elements of various algebraic structures and also the reduced graph defined on the vertices of a compact graph. After introducing and discussing some known results on poset dimension, we will show that poset decomposition can be directly related to the equivalence classes represented in a reduced graph. Using this decomposition, we can build a poset of a compact graph with any dimension in a specified interval. Thus we have a device which gives us the ability to study the dimension of a poset of a zero-divisor graph