Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

The 0—1 Integer Programming Problem in a finite ring with identity

Abstract: We define the 0—1 Integer Programming Problem in a finite field or finite ring with identity as: given an m × n matrix A and an n × 1 vector b with entries in the ring R, find or determine the non-existence of a 0—1 vector x such that Ax = b. We give an easily implemented enumerative algorithm for solving this problem, along with conditions that spurious solutions occur with probability as small as desired. Finally, we show that the problem is NP-complete if R is the ring of integers modulo r for r ≥ 3. This result suggests that it will be difficult to improve on our algorithm.

G-Valued Crystalline Deformation Rings in the Fontaine-Laffaille Range

Abstract: Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf{F}$. Fix a representation $\overline{\rho}$ of the absolute Galois group of an unramified extension of $\mathbf{Q}_p$, valued in $G(\mathbf{F})$. We study the crystalline deformation ring for $\overline{\rho}$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine-Laffaille condition for $G$-valued representations. In particular, we give a root theoretic condition on the $p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.

An isomorphism theorem between the 7-adic integers and a ring associated with a hexagonal lattice

Abstract: The primary goal of this paper is to prove that a ring defined by L Gibson and D Lucas is isomorphic to the ring of 7-adic integers The ring, denoted byR 2, arises naturally as an algebraic structure associated with a hexagonal lattice The elements ofR 2 consist of all infinite sequences in ℤ/(7) The addition and multiplication operations are given in terms of remainder and carries tables The Generalized Balanced Ternary, denoted byG, is the subring ofR 2 consisting of all the finite sequences ofR 2 IfI k ′ is the ideal ofG consisting of all those sequences whose firstk digits are zero, then the second goal of the paper is to show that the inverse limit ofG/I k ′ is also isomorphic to the 7-adic integers

Congruent numbers

Abstract: Number theory is the part of mathematics concerned with the mysterious and hidden properties of the integers and rational numbers (by a rational number, we mean the ratio of two integers). The congruent number problem, the written history of which can be traced back at least a millennium, is the oldest unsolved major problem in number theory, and perhaps in the whole of mathematics. We say that a right-angled triangle is “rational” if all its sides have rational length. A positive integer N is said to be “congruent” if it is the area of a rational right-angled triangle. If we multiply any congruent number N by the square of an integer, we again get a congruent number, and so it suffices to consider only those integers N that are square-free (meaning not divisible by the square of an integer >1). The congruent number problem is simply the question of deciding which square-free positive integers are, or are not, congruent numbers. Long ago, it was realized that an integer N ≥ 1 is congruent if and only if there exists a point (x, y) on the elliptic curve y2 = x3 − N2x, with rational coordinates x, y and with y ≠ 0. Until the 17th century, mathematicians made numerical tables of congruent numbers by using ingenuity to write down the corresponding rational right-angled triangles. For example, the integers 5, 6, and 7 were all known to be congruent, as they are the areas of the right-angled triangles, whose sides lengths are given respectively by [40/6, 9/6, 41/6], [3, 4, 5], and [288/60, 175/60, 337/60]. The first important theoretical result about congruent numbers was established by Fermat, who proved in the 17th century that 1 is not a congruent number. As explained in more …