scholarly journals The congruence speed formula

2021 ◽  
Vol 27 (4) ◽  
pp. 43-61
Author(s):  
Marco Ripà ◽  
Keyword(s):  
Explicit Formula ◽  
Prime Numbers ◽  
Open Problems ◽  
Peculiar Property

We solve a few open problems related to a peculiar property of the integer tetration ^{b}a, which is the constancy of its congruence speed for any sufficiently large b = b(a). Assuming radix-10 (the well known decimal numeral system), we provide an explicit formula for the congruence speed V(a) ∈ ℕ_0 of any a ∈ ℕ − {0} that is not a multiple of 10. In particular, for any given n ∈ ℕ, we prove to be true Ripà’s conjecture on the smallest a such that V(a) = n. Moreover, for any a ≠ 1 ∶ a ≢ 0 (mod 10), we show the existence of infinitely many prime numbers, p_j = p_j(V(a)), such that V(p_j) = V(a).

Index, sub-index and sub-factor of groups with interactions to number theory

2019 ◽  
Vol 19 (06) ◽  
pp. 2050101
Author(s):  
M. H. Hooshmand
Keyword(s):  
Number Theory ◽  
Direct Product ◽  
Prime Numbers ◽  
Additive Number Theory ◽  
Open Problems ◽  
Famous Conjecture ◽  
Future Directions ◽  
Left And Right ◽  
The Difference ◽  
Equivalent Conditions

This paper is the first step of a new topic about groups which has close relations and applications to number theory. Considering the factorization of a group into a direct product of two subsets, and since every subgroup is a left and right factor, we observed that the index conception can be generalized for a class of factors. But, thereafter, we found that every subset [Formula: see text] of a group [Formula: see text] has four related sub-indexes: right, left, upper and lower sub-indexes [Formula: see text], [Formula: see text] which agree with the conception index of subgroups, and all of them are equal if [Formula: see text] is a subgroup or normal sub-semigroup of [Formula: see text]. As a result of the topic, we introduce some equivalent conditions to a famous conjecture for prime numbers (“every even number is the difference of two primes”) that one of them is: the prime numbers set is index stable (i.e. all of its sub-indexes are equal) in integers and [Formula: see text]. Index stable groups (i.e. those whose subsets are all index stable) are a challenging subject of the topic with several results and ideas. Regarding the extension of the theory, we give some methods for evaluation of sub-indexes, by using the left and right differences of subsets. At last, we pose many open problems, questions, a proposal for additive number theory, and show some future directions of researches and projects for the theory.

scholarly journals On trees, tanglegrams, and tangled chains

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
Frderick Matsen IV
Keyword(s):  
Asymptotic Formula ◽  
Computer Science ◽  
Explicit Formula ◽  
Binary Trees ◽  
Biological Research ◽  
Open Problems ◽  
International Audience ◽  
Enumeration Formula ◽  
New Formula ◽  
Simple Asymptotic Formula

International audience Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.

ON THE EXPLICIT FORMULA IN THE THEORY OF PRIME NUMBERS

2012 ◽  
Vol 08 (03) ◽  
pp. 589-597 ◽  
Author(s):  
XIAN-JIN LI
Keyword(s):  
New York ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Li's Criterion

In [Complements to Li's criterion for the Riemann hypothesis, J. Number Theory77 (1999) 274–287] Bombieri and Lagarias observed the remarkable identity [1 - (1 - 1/s)n] + [1 - (1 - 1/(1 - s))n] = [1 - (1 - 1/s)n]⋅[1 - (1 - 1/(1 - s))n], and pointed out that the positivity in Li's criterion [The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory65 (1997) 325–333] has the same meaning as in Weil's criterion [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61]. Let λn = ∑ρ[1 - (1 - 1/ρ)n] for n = 1, 2, …, where ρ runs over the complex zeros of the Riemann zeta function ζ(s). In this note, a certain truncation of λn is expressed as Weil's explicit formula [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61] for each positive integer n. By using the Bombieri and Lagarias' identity, we prove that the positivity of these truncations implies the Riemann hypothesis. If these truncations have suitable upper bounds, we prove that all nontrivial zeros of the Riemann zeta function lie on the critical line.

scholarly journals On the Non-Holonomic Character of Logarithms, Powers, and the $n$th Prime Function

10.37236/1894 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Philippe Flajolet ◽  
Stefan Gerhold ◽  
Bruno Salvy
Keyword(s):  
Differential Equations ◽  
Complex Analysis ◽  
Structure Theorem ◽  
Prime Numbers ◽  
Fractional Powers ◽  
Open Problems ◽  
Naturally Occurring ◽  
Abelian Theorem ◽  
Linear Differential ◽  
Prime Function

We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularity-based methods and several naturally occurring sequences are proved to be non-holonomic.

A remark on a theorem of the Goldbach-Waring type

2004 ◽  
Vol 41 (3) ◽  
pp. 309-324
Author(s):  
C. Bauer
Keyword(s):  
Prime Numbers

Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but ≪ x23027/23040+ε even integers N ≤ x can be written as N = p21 + p32 + p43 + p45.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Continuous Glucose Monitoring Time Series and Hypo/Hyperglycemia Prevention: Requirements, Methods, Open Problems

2008 ◽  
Vol 4 (3) ◽  
pp. 181-192 ◽  
Author(s):  
Giovanni Sparacino ◽  
Andrea Facchinetti ◽  
Alberto Maran ◽  
Claudio Cobelli
Keyword(s):  
Time Series ◽  
Continuous Glucose Monitoring ◽  
Glucose Monitoring ◽  
Open Problems ◽  
Monitoring Time

Secure Cloud Quantum Computing with Verification Based on Quantum Interactive Proof

Impact ◽  
2019 ◽  
Vol 2019 (10) ◽  
pp. 30-32
Author(s):  
Tomoyuki Morimae
Keyword(s):  
Quantum Computing ◽  
Quantum Information ◽  
Theoretical Physics ◽  
View Point ◽  
Research Subjects ◽  
Interactive Proof ◽  
Open Problems ◽  
Main Research ◽  
Proof Systems ◽  
Information Group

In cloud quantum computing, a classical client delegate quantum computing to a remote quantum server. An important property of cloud quantum computing is the verifiability: the client can check the integrity of the server. Whether such a classical verification of quantum computing is possible or not is one of the most important open problems in quantum computing. We tackle this problem from the view point of quantum interactive proof systems. Dr Tomoyuki Morimae is part of the Quantum Information Group at the Yukawa Institute for Theoretical Physics at Kyoto University, Japan. He leads a team which is concerned with two main research subjects: quantum supremacy and the verification of quantum computing.

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