Unsolved Problems in Number Theory

Published on Jan 1, 1981
Richard K. Guy23
Estimated H-index: 23
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Abstract
This monograph contains discussions of hundreds of open questions, organized into 185 different topics. They represent aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In this second edition, new material has been added in addition to corrections.
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This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the problem. It addresses two questions: (1) What can mathematics currently say about this problem? (as of 2010). (2) How can this problem be hard, when it is so easy to state?
Define ||n||to be the complexity of n the smallest number of ones needed to write nusing an arbitrary combination of addition and multiplication. The set \mathscr{D}of defects, differences \delta(n):=||n||-3\log_3 n is known to be a well-ordered subset of [0,\infty) with order type \omega^\omega This is proved by showing that, for any r there is a finite set \mathcal{S}_sof certain multilinear polynomials, called low-defect polynomials, such that \delta(n)\le sif an...
This thesis focuses on two concepts which are widely studied in the field of computational geometry. Namely, visibility and unit disk graphs. In the field of visibility, we have studied the conflict-free chromatic guarding of polygons, for which we have described a polynomial-time algorithm that uses O(n \log^2 n)colors to guard a polygon in a conflict-free setting, and proper coloring of polygon visibility graphs, for which we have described an algorithm that returns a proper 4-coloring for ...
In this paper we find a third order unimodular matrix, none of whose entries is 1or -1 such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular. Further, we find third order square integer matrices (a_{ij}) none of the integers a_{ij}being 1or -1 such that \det{(a_{ij})}=kand \det{(a_{ij}^3)}=k^3 where kis a nonzero integer.
#2Mario Krenn (MPG: Max Planck Society)H-Index: 29
Last. Al谩n Aspuru-GuzikH-Index: 98
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Logic artificial intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and be solved by checking their satisfiability (SAT). Recently, SAT solvers have become a sophisticated and powerful computational tool capable, among other things, of solving long-standing mathematical c...
#3Lukas Spiegelhofer (University of Leoben)H-Index: 7
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#1Alexander Guterman (MSU: Moscow State University)H-Index: 14
#2D. K. Kudryavtsev (MSU: Moscow State University)H-Index: 2
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