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Prime numbers are the integers that cannot be divided exactly by another integer other than one and itself. Prime numbers are notoriously disobedient to rules: they seem to be randomly distributed among natural numbers with no laws except that of chance. Questions about prime numbers have been perplexing mathematicians over centuries. How to efficiently predict greater prime numbers has been a great challenge for many. Most of the previous studies focus on how many prime numbers there are in certain ranges or patterns of the first or last digits of prime numbers. Honestly, although these patterns are true, they help little with accurately predicting new prime numbers, as a deviation at any digit is enough to annihilate the primality of a number. The author demonstrates the periodicity and inter-relationship underlying all prime numbers that makes the occurrence of all prime numbers predictable. This knowledge helps to fish all prime numbers within one net and will help to speed up the related research.

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.

For a natural number $m \ge 2$, we study $m$ layers of finite depth, horizontally infinite, viscous, and incompressible fluid bounded below by a flat rigid bottom. Adjacent layers meet at free interface regions, and the top layer is bounded above by a free boundary as well. A uniform gravitational field, normal to the rigid bottom, acts on the fluid. We assume that the fluid mass densities are strictly decreasing from bottom to top and consider the cases with and without surface tension acting on the free surfaces. In addition to these gravity-capillary effects, we allow a force to act on the bulk and external stress tensors to act on the free interface regions. Both of these additional forces are posited to be in traveling wave form: time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary. Without surface tension in the case of two dimensional fluids and with all positive surface tensions in the higher dimensional cases, we prove that for each sufficiently small force and stress tuple there exists a traveling wave solution. The existence of traveling wave solutions to the one layer configuration ($m=1$) was recently established and, to the best of our knowledge, this paper is the first construction of traveling wave solutions to the incompressible Navier-Stokes equations in the $m$-layer arrangement., 40 pages

Rauzy introduced a fractal set associated with the two-dimensional toric shift by the vector (𝛽−1, 𝛽−2), where 𝛽 is the real root of the equation 𝛽3 = 𝛽2 + 𝛽 + 1 and showed that this fractal is divided into three fractals that are bounded remainder sets with respect to a given toric shift. The introduced set was named as Rauzy fractal. It obtains many applications in the combinatorics of words, geometry, theory of dynamical systems and number theory. Later, an infinite sequence of tilings of 𝑑 − 1-dimensional Rauzy fractals associated with algebraic Pisot units of the degree 𝑑 into fractal sets of 𝑑 types were introduced. Each subsequent tiling is a subdivision of the previous one. These tilings are closely related to some irrational toric shifts and allowed to obtain new examples of bounded remainder sets for these shifts, and also to get some results on self-similarity of shift orbits. In this paper, we continue the study of generalized Rauzy tilings related to Pisot numbers. A new approach to definition of Rauzy fractals and Rauzy tilings based on expansions of natural numbers on linear recurrence sequences is proposed. This allows to improve the results on the connection of Rauzy tilings and bounded remainder sets and to show that the corresponding estimate of the remainder term is independent on the tiling order. The geometrization theorem for linear recurrence sequences is proved. It states that the natural number has a given endpoint of the greedy expansion on the linear recurrence sequence if and only if the corresponding point of the orbit of toric shift belongs to some set, which is the union of the tiles of the Rauzy tiling. Some number-theoretic applications of this result is obtained. In conclusion, some open problems related to generalized Rauzy tilings are formulated.

We prove multilinear variants of the Maurey factorization theorem. Let n be a natural number, π = A 1 , … , A k a partition of the set 1 , … , n , 0 < q 1 , … , q k < ∞ , 1 / v k = 1 / q 1 + ⋯ + 1 ...

The Collatz conjecture establishes that for every natural number n ∈ ℕ, there exists an r ∈ ℕ such that kr(n) = 1, where k : ℕ → ℕ is the function defined as n / 2 if n es even and as 3n + 1 if n is odd. The map k induces a topology 𝜏k on ℕ. We prove that the Collatz conjecture and connectedness of the space (ℕ,𝜏k)imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture is equivalent to the fact that the space (ℕ,𝜏kis path-connected.

Assume that $K$ is a field and $I_{1}\subsetneq ...\subsetneq I_{t}$ is an ascending chain (of length $t$) of ideals in the polynomial ring $K[x_{1},,...,x_{m}]$, for some $m\geq 1$. Suppose that $I_{j}$ is generated by polynomials of degrees less or equal to some natural number $f(j)\geq 1$, for any $j=1,...,t$. In the paper we construct, in an elementary way, a natural number $\mathcal{B}(m,f)$ (depending on $m$ and the function $f$) such that $t\leq\mathcal{B}(m,f)$. We also discuss some possible applications of this result.

Propositional logic, with the aid of SAT solvers, has become capable of solving a range of important and complicated problems. Expanding this range, to contain additional varieties of problems, is subject to the complexity resulting from encoding counting constraints in conjunctive normal form (CNF). Due to the limitation of the expressive power of propositional logic, generally, such an encoding increases the numbers of variables and clauses excessively. This work eliminates the indicated drawback by interpolating constraint symbols and the set of natural numbers $${\mathbb {N}}$$ into the alphabet of propositional logic and adjusting the underlying language accordingly. In the extended logic counting constraints are naturally formulated, while many important aspects, such as Boolean nature and the soundness and completeness theorems, are kept preserved.

This study analysed how preschool teachers differently enacted the same mathematical activity for preschool children to discern numbers, and how this affected the children’s learning opportunities during the activity. The analysis was based on variation theory and Chi’s taxonomy of learning activities. Two Swedish preschool teachers’ enactment of the same mathematical activity for 27 children aged 4–6 years was studied. Video recordings of what the children were offered to discern were used in the analysis. The results indicate that variations in how the teachers chose to enact the activity produced two different learning opportunities for the children. Differences in what aspects were made discernible were closely linked to the characteristics of the activity implemented. The enactments differed even if the same game was chosen and the same amount of time was used in the play-based activity. In one preschool group, there were few opportunities to discern more than the nominal form of numbers; the other preschool group had an activity focused on all number forms simultaneously. In addition, in the latter group, the children had the opportunity to develop equinumerosity. The results suggest that the activity with limited variation was more appropriate for learning with undeveloped knowledge; the children with more developed understanding required a more varied design. This study contributes to the knowledge of how the design of an activity affects children’s learning differently, which is important when planning learning-based preschool activities.

In this paper, we introduce two different generalizations of Schur numbers that involve rainbow colorings. Motivated by well-known generalizations of Ramsey numbers, we first define the rainbow Schur number RS ( n ) to be the minimum number of colors needed such that every coloring of {1, 2, …, n } , in which all available colors are used, contains a rainbow solution to a + b = c . It is shown that RS ( n ) = ⌊log 2 ( n )⌋ + 2, for all n ≥ 3. Second, we consider the Gallai-Schur number GS ( n ) , defined to be the least natural number such that every n -coloring of {1, 2, …, GS ( n )} that lacks rainbow solutions to the equation a + b = c necessarily contains a monochromatic solution to this equation. By connecting this number with the n -color Gallai-Ramsey number for triangles, it is shown that for all n ≥ 3 , GS ( n ) = 5 k if n = 2 k ; GS ( n ) = 2 · 5 k if n = 2 k + 1.

We establish a series of results on verbally closed and l-verbally closed subgroups of free solvable groups; here l is a natural number and the concept of l-verbal closedness is a generalization of the concept of verbal closedness corresponding to the value l = 1. Under certain assumptions, these subgroups turn out to be retracts and, consequently, are algebraically closed.

Purpose The distribution of natural numbers in the Ulam spiral manifests a series of unexpected regularities of the elusive prime numbers. Their sequencing remains a topic of research interest, with many ramifications in different branches of Mathematics, especially in number theory and the prime factorisation problem. Accordingly, the focus of the research is on the most known and widespread asymmetric cryptographic system, that is, the RSA encryption. Design/methodology/approach This paper presents the presence of one, two, three or four adjacencies for the diverse primes that appear in a spiral, considering the Hardy–Littlewood twin prime conjecture and the constellations of primes. Findings Through equations, the calculation formulas of primes inside a spiral that have one to four primes in their adjacent places is offered, with approximate expressions that facilitate the operations, showing that the adjacencies decrease very rapidly as the spiral progresses, although without disappearing. Originality/value A generalisation to cases in which the distances to the prime values change in an ascending way in accordance with the step of the Ulam spiral is offered.

Let $p\ge 5$ be a prime and let $n$ be a natural number. In this article we describe the irreducible constituents of the induced characters $\phi\big\uparrow^{\mathfrak{S}_n}$ for arbitrary linear characters $\phi$ of a Sylow $p$-subgroup of the symmetric group $\mathfrak{S}_n$, generalising earlier results of the authors. By doing so, we introduce Sylow branching coefficients for symmetric groups., Comment: 31 pages

It is shown that there exist arbitrarily large natural numbers $$N$$ and distinct nonnegative integers $$n_1,\dots,n_N$$ for which the number of zeros on $$[-\pi,\pi)$$ of the trigonometric polynomial $$\sum_{j=1}^N \cos(n_j t)$$ is $$O(N^{2/3}\log^{2/3} N)$$ .

Summary This article introduces extended natural numbers, i.e. the set ℕ ∪ {+∞}, in Mizar [4], [3] and formalizes a way to list a cardinal numbers of cardinals. Both concepts have applications in graph theory.

We introduce $\mathsf{LEM}$, a type-assignment system for the linear $ \lambda $-calculus that extends second-order $\mathsf{IMLL}_2$, i.e., intuitionistic multiplicative Linear Logic, by means of logical rules that weaken and contract assumptions, but in a purely linear setting. $\mathsf{LEM}$ enjoys both a mildly weakened cut-elimination, whose computational cost is cubic, and Subject reduction. A translation of $\mathsf{LEM}$ into $\mathsf{IMLL}_2$ exists such that the derivations of the former can exponentially compress the dimension of the derivations in the latter. $\mathsf{LEM}$ allows for a modular and compact representation of boolean circuits, directly encoding the fan-out nodes, by contraction, and disposing garbage, by weakening. It can also represent natural numbers with terms very close to standard Church numerals which, moreover, apply to Hereditarily Finite Permutations, i.e. a group structure that exists inside the linear $ \lambda $-calculus., Comment: 43 pages (10 pages of technical appendix). The final version will appear on Theoretical Computer Science https://doi.org/10.1016/j.tcs.2020.05.001

In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th powers, n=x_1^k+ldots+x_s^k. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. The XIX century problem is still unsolved even or cubes. However, even the solution of the original Waring problem was [almost] finalised only in 1984, with heavy use of computers. In the present paper we document the history of this classical problem itself and its solution, as also discuss possibilities of using this and surrounding material in education, and some further related aspects.

The notion of $\theta $ -congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle $\theta $ such that $\cos \theta $ is a rational number. In this paper we discuss a criterion for a natural number to be $\theta $ -congruent over certain real number fields.

The paper is devoted to the duality of the Mayer problem for κ-th order viable differential inclusions with endpoint constraints, where κ is an arbitrary natural number. Thus, this paper for constr...