## Number Theory

Don H. Tucker and I have been working on the Twin Prime Conjecture for about six or seven years now. We have developed a mathematical algorithm which, when tested using a computer analogue, correctly predicted the twin primes in ascending order up to 5,000,000. Of course, the computer is never a proof (except maybe by intimidation), so we have been working on the induction argument for quite some time. It always seems to be within grasp, and just when I'm about to say, "Oh, to hell with it," I stare back down onto the page and the numbers give me something, they always give me something, something to come back and work on the problem again. Damn! I thought I'd get away!!!!

A Characterization of Twin Prime Pairs,(with Don H. Tucker). Proceedings - Fifth National Conference on Undergraduate Research, Volume I, pp. 362-366, University of North Carolina Press, University of North Carolina at Asheville (UNCA), 1991.AbstractThe basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs(p,p+2)in which bothpandp+2are prime integers.

The basic idea of the arguments is to decompose the integersmodulo 6and then subtract from this set all composites, leaving only primes, then discard further leaving only twin primes pairs.

Remodulization of Congruences,(with Don H. Tucker). Proceedings - Sixth National Conference on Undergraduate Research, Volume II, pp. 1036-1041, University of North Carolina Press, University of North Carolina at Asheville, 1992.AbstractRemodulization introduces a new method applied to congruences and systems of congruences. We prove theChinese Remainder Theoremusing the remodulization method and establish an efficient method to solve linear congruences. The following is an excerpt ofRemodulization of Congruences and its Applications.

Complementary Sets of Systems of Congruences,(with Don H. Tucker). Proceedings - Seventh National Conference on Undergraduate Research, Volume II, pp. 793-796, University of North Carolina Press, University of North Carolina at Asheville, 1993.AbstractWe introduce remodulization and use it to characterize the complementary sets of systems of congruences. The following is an excerpt of a continuing effort to characterize systems of congruences.

On a Conjecture of Erdös,(with Don H. Tucker). Proceedings - Eighth National Conference on Undergraduate Research, Volume II, pp. 794-798, University of North Carolina Press, University of North Carolina at Asheville, 1994.AbstractIn this paper we present some preliminary results on a conjecture by Paul Erdös concerning covering sets of congruences. A covering set consists of a finite system of congruences with distinct moduli, such that every integer satisfies as a minimum one of the congruences. An interesting consequence of this conjecture is the dependence of the solution on abundant numbers; an abundant number is an integer whose sum of its proper divisors exceeds the integer.

A Novel Solution of Linear Congruences,(with Don H. Tucker). Proceedings - Ninth National Conference on Undergraduate Research, Volume II, pp. 708-712, University of North Carolina Press, University of North Carolina at Asheville, 1995.AbstractAlthough the solutions of linear congruences have been of interest for a very long time, they still remain somewhat pedagogically difficult. Because of the importance of linear congruences in fields such as public-key cryptosystems, new and innovative approaches are needed both to attract interest and to make them more accessible. While the potential for new ideas used in future research is difficult to assess, some use may be found here.

In this paper, the authors make use of the remodulization method developed in a previous paper as a vehicle to characterize the conditions under which solutions exist and then determine the solution space. The method is more efficient than those cited in the standard references. This novel approach relates the solution space ofcx = a mod bto the Euler totient function forcrather than that ofb, which allows one to develop an alternative and somewhat more efficient approach to the problem of creating enciphering and deciphering keys in public-key cryptosystems.

Vector Products Revisited: A New and Efficient Method of Proving Vector Identities,(with Don H. Tucker). Proceedings - Tenth National Conference on Undergraduate Research, Volume II, University of North Carolina Press, University of North Carolina at Asheville, 1996. To appear.AbstractThe purpose of these remarks is to introduce avariation on a themeof the scalar (inner, dot) product and establish multiplication inR^{n}. Ifa= (a_{1},...,a_{n}) andb=(b_{1},...,b_{n}), we define the productab= (a_{1}b_{1},...,a_{n}b_{n}). The product is the multiplication of corresponding vector components as in the scalar product; however, instead of summing the vector components, the product preserves them in vector form. We define the inner sum (or trace) of a vectora= (a_{1},...,a_{n}) bys(a)=a_{1}+...+a_{n}. If taken together with an additional definition of cyclic permutations of a vector^{[p]}a= (a_{1+p (mod n)},...,a_{n+p (mod n)}),where

ainR^{n}and the permutation exponentpinZ, we are able to prove complicated vector products (combinations of dot and cross products) extremely efficiently, without appealing to the traditional (and cumbersome)epsilon-ijkproofs. When applied to determinants, this method hints at the rudiments ofGaloistheory.