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Characterizing The Properties Of Specific Binomial Coefficients In Congruence Relations, Tyler Robert Russ
Characterizing The Properties Of Specific Binomial Coefficients In Congruence Relations, Tyler Robert Russ
Master's Theses and Doctoral Dissertations
The number theoretic conjecture we examine in this paper originates when trying to construct a characterizable generating set for the complex cobordism polynomial ring. To date there is no efficient, universal method for characterizing such a generating set. Wilfong conjectures that smooth projective toric varieties can act as these generators [7]. Toric varieties are related to polytopes by a bijective correspondence. Studying the combinatorial structure of these polytopes is much more manageable than studying properties of toric varieties directly. This gives rise to the number theoretic conjecture considered here. A proof of this number theoretic conjecture would in turn prove …
A Short Note On Sums Of Powers Of Reciprocals Of Polygonal Numbers, Jihang Wang, Suman Balasubramanian
A Short Note On Sums Of Powers Of Reciprocals Of Polygonal Numbers, Jihang Wang, Suman Balasubramanian
Student Research
This paper presents the summation of powers of reciprocals of polygonal numbers. Several summation formulas of the reciprocals of generalized polygonal numbers are presented as examples of specific cases in this paper.
Commutative N-Ary Arithmetic, Aram Bingham
Commutative N-Ary Arithmetic, Aram Bingham
University of New Orleans Theses and Dissertations
Motivated by primality and integer factorization, this thesis introduces generalizations of standard binary multiplication to commutative n-ary operations based upon geometric construction and representation. This class of operations are constructed to preserve commutativity and identity so that binary multiplication is included as a special case, in order to preserve relationships with ordinary multiplicative number theory. This leads to a study of their expression in terms of elementary symmetric polynomials, and connections are made to results from the theory of polyadic (n-ary) groups. Higher order operations yield wider factorization and representation possibilities which correspond to reductions in the set of primes …
Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr
Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr
Honors College Theses
The Collatz Conjecture is a deceptively difficult problem recently developed in mathematics. In full, the conjecture states: Begin with any positive integer and generate a sequence as follows: If a number is even, divide it by two. Else, multiply by three and add one. Repetition of this process will eventually reach the value 1. Proof or disproof of this seemingly simple conjecture have remained elusive. However, it is known that if the generated Collatz Sequence reaches a cycle other than 4, 2, 1, the conjecture is disproven. This fact has motivated our search for occurrences of 4, 2, 1, and …
Integral Generalized Binomial Coefficients Of Multiplicative Functions, Imanuel Chen
Integral Generalized Binomial Coefficients Of Multiplicative Functions, Imanuel Chen
Summer Research
The binomial coefficients are interestingly always integral. However, when you generalize the binomial coefficients to any class of function, this is not always the case. Multiplicative functions satisfy the properties: f(ab) = f(a)f(b) when a and b are relatively prime, and f(1) = 1. Tom Edgar of Pacific Lutheran University and Michael Spivey of the University of Puget Sound developed a Corollary that determines which values of n and m will always have integral generalized binomial coefficients for all multiplicative functions. The purpose of this research was to determine as many patterns within this corollary as possible as well as …
On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh
On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh
CMC Senior Theses
This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.
Natural Neutrosophic Numbers And Mod Neutrosophic Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral
Natural Neutrosophic Numbers And Mod Neutrosophic Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral
Branch Mathematics and Statistics Faculty and Staff Publications
In this book authors answer the question proposed by Florentin Smarandache “Does there exist neutrosophic numbers which are such that they take values differently and behave differently from I; the indeterminate?”. We have constructed a class of natural neutrosophic numbers m 0I , m xI , m yI , m zI where m 0I × m 0I = m 0I , m xI × m xI = m xI and m yI × m yI = m yI and m yI × m xI = m 0I and m zI × m zI = m 0I . Here take m …