Sequences of integers generated by two fixed primes

Good sequences of integers

Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n (ξ) and w n * (ξ) defined by Mahler and Koksma. We calculate their six values when n=2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction ξ by quadratic surds.

We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called k-LCIS: Given k integer sequences X_1,dots ,X_k of length at most n, the task is to determine the length of the longest common subsequence of X_1,dots ,X_k that is also strictly increasing. Especially for the case of k=2 (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case. Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as Longest Common Subsequence. We further strengthen this lower bound (1) to rule out {mathcal {O}}left( (nL)^{1-varepsilon }right) time algorithms for LCIS, where L denotes the solution size, (2) to rule out {mathcal {O}}left( n^{k-varepsilon }right) time algorithms for k-LCIS, and (3) to follow already from weaker variants of SETH. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.

RSA cryptanalysis — Fermat factorization exact bound and the role of integer sequences in factorization problem

<abstract><p>In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.</p></abstract>

Unique expansions of real numbers

The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} \beta^{-w(m)}\), where \(\beta\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with \(w(m+1)>w(m)\) for any sufficiently large \(m\). We first introduce the algebraic independence results of such values. Our results are applicable to certain sequences \(w(m)\) (\(m=0,1,\ldots\)) with \(\lim_{m\to\infty}w(m+1)/w(m)=1.\) For example, we prove that two numbers \[\sum_{m=1}^{\infty}\beta^{-\lfloor \varphi(m)\rfloor}, \quad \sum_{m=3}^{\infty}\beta^{-\lfloor a(m)\rfloor}\] are algebraically independent, where \(\varphi(m)=m^{\log m}\) and \(a(m)=m^{\log\log m}\). Moreover, we also give the linear independence results of real numbers. Our results are applicable to the values \(\sum_{m=0}^{\infty}\beta^{-\lfloor m^\rho\rfloor}\), where \(\beta\) is a Pisot or Salem number and \(\rho\) is a real number greater than 1.

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$ . In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull.20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$ . Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar.160 (2020), 360–375] improved Borwein’s upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$ . In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained.

On strings containing all subsets as substrings

Part 1 The Real Numbers: The Real Number System Upper and Lower Bounds. Part 2 Sequences and Series: Algebraic Operations on Limits Monotone Sequences Infinite Series. Part 3 Continuous Functions: Functions, Limits and Continuity The Intermediate Value Property for Continuous Functions Uniform Continuity Increasing Functions. Part 4 Differentiable Functions: Repeated Differentiation Mean Value Theorems Local Maxima and Minima Taylor's Theorem. Part 5 Further Results on Infinite Series: Tests for Convergence Series of Complex Terms Power Series Multiplication of Series. Part 6 Special Functions: The Exponential Function The Logarithm Trigonometric Functions Inverse Trigonometric Functions. Part 7 The Riemann Integral: Integral Forms of the Mean Value Theorems Integration Over Unbounded Intervals Integration of Unbounded Functions. Part 8 The Number Pi.

The three-distance theorem (also known as the three-gap theorem or Steinhaus problem) states that, for any given real number $\alpha $ and integer $N$, there are at most three values for the distances between consecutive elements of the Kronecker sequence $\alpha , 2\alpha ,\ldots , N\alpha $ mod 1. In this paper, we consider a natural generalization of the three-distance theorem to the higher-dimensional Kronecker sequence $\vec \alpha , 2\vec \alpha ,\ldots , N\vec \alpha $ modulo an integer lattice. We prove that in 2D, there are at most five values that can arise as a distance between nearest neighbors, for all choices of $\vec \alpha $ and $N$. Furthermore, for almost every $\vec \alpha $, five distinct distances indeed appear for infinitely many $N$ and hence five is the best possible general upper bound. In higher dimensions, we have similar explicit, but less precise, upper bounds. For instance, in 3D, our bound is 13, though we conjecture the truth to be 9. We furthermore study the number of possible distances from a point to its nearest neighbor in a restricted cone of directions. This may be viewed as a generalization of the gap length in 1D. For large cone angles, we use geometric arguments to produce explicit bounds directly analogous to the three-distance theorem. For small cone angles, we use ergodic theory of homogeneous flows in the space of unimodular lattices to show that the number of distinct lengths is (1) unbounded for almost all $\vec \alpha $ and (2) bounded for $\vec \alpha $ that satisfy certain Diophantine conditions.

In this paper, we bring together two concepts related to uncertainty and vagueness: fuzzy numbers and intervals. With them, we build a new structure whose elements we call interval fuzzy segments. We have undertaken this based on the conviction that the fuzzy numbers are a correct representation of the real numbers under situations of indeterminacy. We also believe that if it makes sense to consider the set of real numbers between two real bounds, then it also makes sense to consider the set of all the fuzzy numbers between two fuzzy number bounds. In this way, we extend the concept of real interval to the concept of interval fuzzy segment defined by two fuzzy bounds and a transition mapping that leads from the lower fuzzy bound to the upper fuzzy bound and this transition mapping generates the set of all the fuzzy numbers comprised between those fuzzy bounds. At the same time, this transition mapping brings the concept of interval fuzzy segment closer to the concept of line segment.

The communication cost of algorithms (also known as I/O-complexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen's and other fast matrix multiplication algorithms, and obtain first lower bounds on their communication costs. In the sequential case, where the processor has a fast memory of size $M$, too small to store three $n$-by-$n$ matrices, the lower bound on the number of words moved between fast and slow memory is, for many of the matrix multiplication algorithms, $\Omega((\frac{n}{\sqrt M})^{\omega_0}\cdot M)$, where $\omega_0$ is the exponent in the arithmetic count (e.g., $\omega_0 = \lg 7$ for Strassen, and $\omega_0 = 3$ for conventional matrix multiplication). With $p$ parallel processors, each with fast memory of size $M$, the lower bound is $p$ times smaller. These bounds are attainable both for sequential and for parallel algorithms and hence optimal. These bounds can also be attained by many fast algorithms in linear algebra (e.g., algorithms for LU, QR, and solving the Sylvester equation).

Consider the difference equation oo (E) (-l)"+,A"4, + ;[>4,-4-0.k=0 where m is a positive integer, (Pk)k>o is a sequence of positive real numbers and (/(t)/t>o 's a sequence of integers with 0 < o < l\ < 12 < .The characteristic equation of (E) is CO (*) -(i-Ar + x>_/* = o.*=0We prove the following theorem.Theorem, (i) For m even, (E) has a positive solution (An)nez w'th limsup"_00 A" < oo if and only if (*) has a root in (0, 1).(ii) For m odd, (E) has a positive solution (A")nez if and only if (*) has a root in (0,1).

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor &gt; 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2N: (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (Р1-Р2-Р3). In this case, the distance from the first to the third prime number of any group is less than 2N integers, that is, Р3–Р1 &lt; 2N integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than 2Nintegers.