Tilings in the 3 dimensional lattice with L-tetrominoes

We consider three dimensional L-tetrominoes. We show that there exists at least one way to tile every three dimensional rectangle whose side lengths are at least $3$ and area is congruent to $1 \pmod 4$ such that one square goes untiled. In addition, we show that every three dimensional rectangle is tileable provided one side has length at least $2$ and the other is a multiple of $4$

Incremental Increases Between Successive Integers when Raised to the nth Power

For thousands of years, the beautiful field of number theory has captivated mathematicians with its elegant simplicity. Positive integers continue to reveal properties and relationships that are a joy to uncover, and in this paper, we investigate a pattern involving exponents and factorials while exploring some common notations in the field of number theory. Combinatorics, the field dealing with the mathematics of counting and arranging, also holds a presence in this paper. Pascal’s Triangle–the foundation of binomial expressions, also comes into play due to its tight relationship with combinatorics. Pascal’s Identity, the property that builds the triangle, becomes very useful as well. In our study, we explore the incremental increases between successive integers when raised to the nth power. For example, if we raise consecutive positive integers to the second power and enact two orders of differences on these values, we arrive at the constant increment of two, which is 2!. Further, if raise consecutive positive integers to the third power and enact three orders of differences, we obtain the constant increment of six–which is 3!. In this paper, we prove that if we raise consecutive positive integers to the nth power and take the nth difference, we always arrive at the constant increment of n!

Counting Rotational Sets for Laminations of the Unit Disk from First Principles

By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, d, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of rotational sets\u27\u27, which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure of these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials. These rotational sets are created by uniting rotational orbits, as we define in this paper. The number of such sets for a given degree d, rotation number p/q, and cardinality k can be determined by analyzing the potential placements of pre-images of zero on the unit circle with respect to the rotational set under the d-tupling map. We obtain a closed-form formula for the count. Though this count is already known based upon some sophisticated results, our count is based upon elementary geometric and combinatorial principles, and provides an intuitive explanation

How Many Symmetries of the Regular n-gon Are Even?

One of the simplest classes of finite groups used as a source of counterexamples in a first course of modern algebra is the class of finite dihedral groups. Among the subgroups of dihedral group, finding subgroups of index 2 is of interest in part because these subgroups are normal subgroups. In this article, we use the representations of the symmetries of the dihedral groups as permutations of the vertices and determine concretely all its subgroups of index 2. Under this representation or embedding, the article determines the intersection of the dihedral group with the corresponding alternating groups when they are naturally viewed as subgroup of the symmetry group on the set of vertices of the regular n-gon. In its second part, the article considers the question of embedding an arbitrary finite group into a symmetry group using the well-known Cayley embedding. In this more general context where every element of the group is viewed as a permutation, one counts the even elements of the group

Finite Posets as Prime Spectra of Commutative Noetherian Rings

We study finite partially ordered sets of prime ideals as found in commutative Noetherian rings. In doing so, we establish that these posets have a bipartite structure and devise a construction for finding ring spectra that are order-isomorphic to many such posets. Specifically, we prove that any finite complete bipartite graph is order-isomorphic to the spectrum of a ring of essentially finite type over the field of rational numbers. Furthermore, we prove that prime spectra of such rings can also depict any finite path or even cycle

The Intricacies of Pairwise Modular Multiplicative Inverse in Lucas Numbers

Let (p,q) be a pair of relatively prime integers greater than 1. The pairwise modular multiplicative inverse (PMMI) of (p,q) is defined as the unique pair of positive integers (p′, q′) such that p p′ ≡ 1 (mod q), p′ \u3c q, qq′ ≡ 1 (mod p), q′ \u3c p. In this paper, we determine all pairs of Lucas numbers such that their PMMIs are pairs of Lucas numbers