arXiv subjects

Benjamin V. Holt

Publications and source records attributed to Benjamin V. Holt.

10 recordsLinked to original sources

arXiv2026

Producing Quality Pseudorandomness with a Generalized Gauss Continued-Fraction Map

Well-known chaotic maps, such as the logistic and tent maps, have been used to generate cryptographically secure pseudorandomness, yet we know of no efforts which attempt to utilize the Gauss continued-fraction map, a known chaotic map, as a starting point for producing quality pseudorandom output. In this paper, we consider the family of $r$-continued-fraction maps, which generalize the Gauss map, and use them to generate pseudorandom output which outperforms many standard generators, such as the Mersenne Twister, in statistical quality, as ascertained by the use of the Dieharder, PractRand, and TestU01 suites. In this way, we demonstrate the potential viability of these maps as a starting point for novel generators, and provide practical motivation for further study of the properties of both the exact and finite-precision $r$-continued-fraction maps.

math.DS↗

arXiv2025

A Multigraph Characterization of Permutiple Strings

A permutiple is a natural number whose representation in some base is an integer multiple of a number whose representation has the same collection of digits. A previous paper utilizes a finite-state-machine construction and its state graph to recognize permutiples and to generate new examples. Permutiples are associated with walks on the state graph which necessarily satisfy certain conditions. However, the above effort does not provide sufficient conditions for the existence of permutiples. In this paper, we provide such a condition which we will state using the language of multigraphs.

math.CO↗

arXiv2025

Utilizing Symmetry in Finding New Permutiples from Known Examples

A permutiple is a natural number whose representation in some base, $b>1$, is an integer multiple of a number whose base-$b$ representation has the same collection of digits. Previous efforts have made progress in finding such numbers using graph-theoretical and finite-state-machine constructions. These are the mother graph and the Hoey-Sloane machine. In this paper, we leverage the inherent symmetry of the above constructions for the purpose of finding new permutiples from old. Such results also help us to see previous work through a new lens.

math.CO↗

arXiv2025

Finding Permutiples of a Known Base and Multiplier

Natural numbers which are nontrivial multiples of some permutation of their base-$b$ digit representations are called permutiples. Specific cases include numbers which are multiples of cyclic permutations (cyclic numbers) and reversals of their digits (palintiples). Previous efforts have produced methods which construct new examples of permutiples with the same set of digits as a known example. Using simple graph-theoretical and finite-state machine constructions, we advance previous work by describing two methods for finding permutiples of a known base and multiplier with no need for known examples or prior knowledge of digits.

math.CO↗

arXiv2025

A Method for Finding All Permutiples with a Fixed Set of Digits from a Single Known Example

A permutiple is a natural number that is a nontrivial multiple of a permutation of its digits in some base. Special cases of permutiples include cyclic numbers (multiples of cyclic permutations of their digits) and palintiple numbers (multiples of their digit reversals). While cyclic numbers have a fairly straightforward description, palintiple numbers admit many varieties and cases. A previous paper attempts to get a better handle on the general case by constructing new examples of permutiples with the same set of digits, multiplier, and length as a known example. However, the results are not sufficient for finding all possible examples except when the multiplier divides the base. Using an approach based on the methods of this previous paper, we develop a new method which enables us to find all examples under any conditions.

math.NT↗

arXiv2017

Derived Palintiple Families and Their Palinomials

We consider several families of palintiples (also known as reverse multiples) whose carries themselves are digits of lower-base palintiples and give some methods for constructing them from fundamental palintiple types. We also continue the study of palinomials introduced in an earlier paper by revealing a more direct relationship between the digits of certain palintiple types and the roots of their palinomials. We explore the consequences of this relationship for palinomials induced by palintiple families derived from lower-base palintiples. Finally, we pose some questions regarding Young graphs of derived palintiple families and consider the implications our general observations might have for relations between Young graph isomorphism classes.

math.NT↗

arXiv2017

Digit Preserving Multiplication in Continued Fraction Representations

A permutiple is a number which is an integer multiple of some permutation of its digits. A well-known example is 9801 since it is an integer multiple of its reversal, 1089. In this paper, we consider the permutiple problem in an entirely different setting: continued fractions. We pose the question of when the simple continued fraction representation of a rational number is an integer multiple of a permutation of its partial quotients (or digits, as we shall call them). We develop some general results and apply them to finding new examples. In doing so, we attempt to classify all 2, 3, and 4-digit continued fraction permutiples in terms of basic permutiple types which we discover along the way. We also generate new examples from old by finding conditions which guarantee that digit-string concatenation yields other permutiples.

math.NT↗

arXiv2017

On Permutiples Having a Fixed Set of Digits

A permutiple is the product of a digit preserving multiplication, that is, a number which is an integer multiple of some permutation of its digits. Certain permutiple problems, particularly transposable, cyclic, and, more recently, palintiple numbers, have been well-studied. In this paper we study the problem of general digit preserving multiplication. We show how the digits and carries of a permutiple are related and utilize these relationships to develop methods for finding new permutiple examples from old. In particular, we shall focus on the problem of finding new permutiples from a known example having the same set of digits.

math.NT↗

arXiv2015

Some Thoughts on Determining Symmetric Palintiples

A palintiple is a natural number which is an integer multiple of its digit reversal. A previous paper partitions all palintiples into three distinct classes according to patterns in the carries and then determines all palintiples belonging to the shifted-symmetric class. In this paper we consider a strategy for finding all symmetric palintiples based upon recent work. We also discuss the last case of asymmetric palintiples and consider bases for which asymmetric palintiples do not exist.

math.NT↗

arXiv2005

Group actions in number theory

Students having had a semester course in abstract algebra are exposed to the elegant way in which finite group theory leads to proofs of familiar facts in elementary number theory. In this note we offer two examples of such group theoretical proofs using the action of a group on a set. The first is Fermat's little theorem and the second concerns a well known identity involving the famous Euler phi function. The tools that we use to establish both results are sometimes seen in a second semester algebra course in which group actions are studied. Specifically, we will use the class equation of a group action and Burnside's theorem.

math.HO↗