Set Of Prime Numbers Research Articles

There are at most finitely many singular moduli that are S-units

We show that for every finite set of prime numbers $S$ , there are at most finitely many singular moduli that are $S$ -units. The key new ingredient is that for every prime number $p$ , singular moduli are $p$ -adically disperse. We prove analogous results for the Weber modular functions, the $\lambda$ -invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.

TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CURVES IV: DISCRETENESS AND SECTIONS

Abstract Let $\Sigma $ be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro- $\Sigma $ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of $\Sigma $ are invertible. The present paper focuses on the topic of comparison between the theory developed in earlier papers concerning pro- $\Sigma $ fundamental groups and various discrete versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the section conjecture and Grothendieck conjecture. This portion of the theory is purely combinatorial and essentially follows from a result concerning the existence of fixed points of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to scheme theory. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of (discrete) subgroups of such groups hold if and only if analogous properties hold for the closures of these subgroups in the profinite completions of the discrete fundamental groups under consideration. These results make possible a fairly straightforward translation, into discrete versions, of pro- $\Sigma $ results obtained in previous papers by the authors. Finally, we discuss a construction that was considered previously by M. Boggi in the discrete case from the point of view of the present paper.

Finite groups with modular 𝜎-subnormal subgroups

Abstract Let 𝜎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 𝜎-subnormal subgroup is modular.

On simply normal numbers with digit dependencies

AbstractGiven an integer and a set of prime numbers, the set of Toeplitz numbers comprises all elements of [0, b[ whose digits in the base‐b expansion satisfy for all and . Using a completely additive arithmetical function, we construct a number in that is simply Borel normal if, and only if, . We then provide an effective bound for the discrepancy.

“Criticality” in the counting function of prime numbers: Theory and application to nuclear magic numbers

In this paper, considering critical phenomena and their phase transitions within the frame of the set of prime numbers, is attempted. Thus, the novel, theoretical and purely mathematical Model of Criticality based on Prime Numbers is introduced. This approach allows for the emergence of a parallelism between the physical concept of criticality and the corresponding concept in prime number theory. Based on this parallelism an application of the proposed model in determining the known magic numbers of Nuclear Physics is presented. This application introduces a physical meaning to the exceptional in properties set of the prime numbers and the corresponding prime number theory. Finally, going beyond the proposed model and its application, we suggest investigating other prime numbers as doubly magic numbers or candidate magic numbers for further experimental research.

On the separability of subgroups of nilpotent groups by root classes of groups

Abstract Suppose that 𝒞 is a class of groups consisting only of periodic groups and P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} is the set of prime numbers that do not divide the order of any element of a 𝒞-group. It is easy to see that if a subgroup 𝑌 of a group 𝑋 is 𝒞-separable in this group, then it is P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated in 𝑋. Let us say that 𝑋 has the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} if all of its P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated subgroups are 𝒞-separable. We find a condition that is sufficient for a nilpotent group 𝑁 to have the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} provided 𝒞 is a root class. We also prove that if 𝑁 is torsion-free, then the indicated condition is necessary for this group to have C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} .

On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

Let $\pi$ be a proper subset of the set of prime numbers. Denote by $r$ the least prime not contained in $\pi$ and set $m=r$ for $r=2$ and $3$ and $m=r-1$ for $r\geqslant5$. The conjecture under consideration claims that a conjugacy class $D$ of a finite group $G$ generates a $\pi$-subgroup of $G$ (equivalently, is contained in the $\pi$-radical) if and only if any $m$ elements of $D$ generate a $\pi$-group. It is shown that this conjecture holds if every non-Abelian composition factor of $G$ is isomorphic to a sporadic, an alternating, a linear, or a unitary simple group. Bibliography: 49 titles.

A proof of the Erdős primitive set conjecture

Abstract A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum _{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative. As further applications of the method, we make progress towards a question of Erdős, Sárközy and Szemerédi from 1968. We also refine the classical Davenport–Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy and Szemerédi from 1966.

Perfectly packing a square by squares of sidelength f(n)−t

In this paper, we prove that for any 1/2<t<1, there exists a positive integer N0 depending on t such that for any n0≥N0, squares of sidelength f(n)−t for n≥n0 can be packed with disjoint interiors into a square of area ∑n=n0∞f(n)−2t, if the function f satisfies some suitable conditions. The main theorem (Theorem 1.1) is a generalization of Tao's theorem in [15], which argued the case f(n)=n. As corollaries, we prove that there are such packings of squares when f(n) represents the nth element of either an arithmetic progression or the set of prime numbers. In these cases, we give effective lower bounds for N0 with respect to t. Furthermore, we consider the case that f(n) represents the nth element of the set of twin primes and prove that squares of sidelength f(n)−t for n≥n0 can be packed with disjoint interiors into a slightly larger square than theoretically expected.

Prime Representing Polynomial with 10 Unknowns – Introduction. Part II

Summary In our previous work [7] we prove that the set of prime numbers is diophantine using the 26-variable polynomial proposed in [4]. In this paper, we focus on the reduction of the number of variables to 10 and it is the smallest variables number known today [5], [10]. Using the Mizar [3], [2] system, we formalize the first step in this direction by proving Theorem 1 [5] formulated as follows: Let k ∈ ℕ. Then k is prime if and only if there exists f, i, j, m, u ∈ ℕ+, r, s, t ∈ ℕ unknowns such that D F I is square ∧ ( M 2 - 1 ) S 2 + 1 is square ∧ ( ( M U ) 2 - 1 ) T 2 + 1 is square ∧ ( 4 f 2 - 1 ) ( r - m S T U ) 2 + 4 u 2 S 2 T 2 &lt; 8 f u S T ( r - m S T U ) F L | ( H - C ) Z + F ( f + 1 ) Q + F ( k + 1 ) ( ( W 2 - 1 ) S u - W 2 u 2 + 1 ) \matrix{ {DFI\,is\,square\,\,\,{\Lambda}\,\left( {{M^2} - 1} \right){S^2} + 1\,\,is\,\,square\,\,{\Lambda}} \hfill \cr {\left( {{{\left( {MU} \right)}^2} - 1} \right){T^2} + 1\,\,is\,\,square{\Lambda}} \hfill \cr {\left( {4{f^2} - 1} \right){{\left( {r - mSTU} \right)}^2} + 4{u^2}{S^2}{T^2} &lt; 8fuST\left( {r - mSTU} \right)} \hfill \cr {FL|\left( {H - C} \right)Z + F\left( {f + 1} \right)Q + F\left( {k + 1} \right)\left( {\left( {{W^2} - 1} \right)Su - {W^2}{u^2} + 1} \right)} \hfill \cr } where auxiliary variables A − I, L, M, S − W, Q ∈ ℤ are simply abbreviations defined as follows W = 100fk(k + 1), U = 100u 3 W 3 + 1, M = 100mUW + 1, S = (M −1)s+k+1, T = (MU −1)t+W −k+1, Q = 2MW −W 2 −1, L = (k+1)Q, A = M(U +1), B = W +1, C = r +W +1, D = (A 2 −1)C 2 +1, E = 2iC 2 LD, F = (A 2 −1)E 2 +1, G = A+F (F −A), H = B+2(j −1)C, I = (G 2 −1)H 2 +1. It is easily see that (0.1) uses 8 unknowns explicitly along with five implicit one for each diophantine relationship: is square, inequality, and divisibility. Together with k this gives a total of 14 variables. This work has been partially presented in [8].

Computational Analysis of Quantitative Characteristics of Some Residual Properties of Solvable Baumslag–Solitar Groups

Let Gk be defined as Gk = \(\left\langle {a,b;{{a}^{{ - 1}}}ba = {{b}^{k}}} \right\rangle \), where k ≠ 0. It is known that if p is a prime number then Gk is residually a finite p-group iff p|k – 1. It is also known that if p and q are primes not dividing k – 1, p < q and π = {p, q} then Gk is residually a finite π-group iff (k, q) = 1, p|q – 1 and the order of k in the multiplicative group of the field ℤq is a p-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let fk(x) be the number of sets {p, q} such that p < q, p \(\nmid \) k – 1, q \(\nmid \) k – 1, (k, q) = 1, p|q – 1, the order of k modulo q is a p-number, and p and q are chosen among the first x primes. We state that, if 2 ≤ |k| ≤ 10 000 and 1 ≤ x ≤ 50 000, then for almost all considered k the function fk(x) can be approximated quite accurately by the function αkx0.85, where the coefficient αk is different for each k and {αk|2 ≤ |k| ≤ 10 000} ⊆ (0.28; 0.31]. In addition, the dependence of the value fk(50 000) on k is investigated and an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion is proposed. The results may have applications in the theory of computational complexity and algebraic cryptography.

The Green-Tao Theorem and the Infinitude of Primes in Domains

We first prove an elementary analogue of the Green-Tao Theorem. The celebrated Green-Tao Theorem states that there are arbitrarily long arithmetic progressions in the set of prime numbers. In fact, we show the Green-Tao Theorem for polynomial rings over integral domains with several variables. Using the Generalized Polynomial van der Waerden Theorem, we also prove that in an infinite unique factorization domain, if the cardinality of the set of units is strictly less than that of the domain, then there are infinitely many prime elements. Moreover, we deduce the infinitude of prime numbers in the positive integers using polynomial progressions of length three. In addition, using unit equations, we provide two more proofs of the infinitude of prime numbers. Finally, we give a new proof of the divergence of the sum of reciprocals of all prime numbers.

Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups

Let $\Sigma$ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the geometry of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be reconstructed group-theoretically from the pro-$\Sigma$ fundamental group of the configuration space. Let $X$ be a hyperbolic curve of type $(g,r)$ over a field $k$ of characteristic zero. Thus, $X$ is obtained by removing from a proper smooth curve of genus $g$ over $k$ a closed subscheme [i.e., the "divisor of cusps"] of $X$ whose structure morphism to Spec$(k)$ is finite étale of degree $r$; $2g-2+r>0$. Write $X_n$ for the $n$-th configuration space associated to $X$, i.e., the complement of the various diagonal divisors in the fiber product over $k$ of $n$ copies of $X$. Then, when $k$ is algebraically closed, we show that the triple $(n,g,r)$ and the generalized fiber subgroups—i.e., the subgroups that arise from the various natural morphisms $X_n \to X_m$ $[m$ < $n]$, which we refer to as generalized projection morphisms—of the pro-$\Sigma$ fundamental group $\Pi_n$ of $X_n$ may be reconstructed group-theoretically from $\Pi_n$ whenever $n \geq 2$. This result generalizes results obtained previously by the first and third authors and A. Tamagawa to the case of arbitrary hyperbolic curves [i.e., without restrictions on $(g,r)$]. As an application, in the case where $(g,r)= (0,3)$ and $n \geq 2$, we conclude that there exists a direct product decomposition Out$(\Pi_n) = {\mathrm{GT}^{\Sigma}} \times \mathfrak{S}_{n + 3}$ —where we write "Out$(-)$" for the group of outer automorphisms [i.e., without any auxiliary restrictions!] of the profinite group in parentheses and ${\mathrm{GT}^{\Sigma}}$ (respectively, $\mathfrak{S}_{n + 3}$) for the pro-$\Sigma$ Grothendieck-Teichmüller group (respectively, symmetric group on $n + 3$ letters). This direct product decomposition may be applied to obtain a simplified purely group-theoretic equivalent definition—i.e., as the centralizer in ${\rm Out}(\Pi_n)$ of the union of the centers of the open subgroups of ${\rm Out}(\Pi_n)$—of ${\mathrm{GT}^{\Sigma}}$. One of the key notions underlying the theory of the present paper is the notion of a pro-$\Sigma$ log-full subgroup—which may be regarded as a sort of higher-dimensional analogue of the notion of a pro-$\Sigma$ cuspidal inertia subgroup of a surface group—of $\Pi_n$. In the final section of the present paper, we show that, when $X$ and $k$ satisfy certain conditions concerning "weights", the pro-$l$ log-full subgroups may be reconstructed group-theoretically from the natural outer action of the absolute Galois group of $k$ on the geometric pro-$l$ fundamental group of $X_n$.

Bell numbers and Kurepa’s conjecture

We prove under a mild condition that Kurepa's conjecture holds for the set of prime numbers \(p\) such that \((\frac{p-1}{2})! = {2 \overwithdelims () p\;}\) in \(\mathbb{F}_p\).

SPACINGS BETWEEN SELECTED PRIME DIVISORS OF AN INTEGER

AbstractGiven a large integer n, determining the relative size of each of its prime divisors as well as the spacings between these prime divisors has been the focus of several studies. Here, we examine the spacings between particular types of prime divisors of n, such as prime divisors in certain congruence classes of primes and various other subsets of the set of prime numbers.

Note on the natural density of r-free numbers

Let P be a finite set of prime numbers. By using an elementary method, the proportion of all r-free numbers which are divisible by at least one element in P is studied.

Alternative Representations of Some Arithmetic Functions

This article presents some results of the attempt to simplify the writing of arithmetic functions on the computer so that users can apply them without additional operations, such as summing after a set whose elements must be calculated, such as the set of numbers prime. The important role of the remainder function in defining most arithmetic functions is highlighted. Defining algorithms for the prime factorization of natural numbers highlights the possibility of representing natural numbers in a basis as "natural" as possible for natural numbers, namely the basis of prime numbers. The disadvantage of this natural basis is, for the time being, that it is infinitely dimensional. For now, this representation provides advantages but also disadvantages. Among the arithmetic functions proposed in the article, there are also statistical characterizations of the distribution of prime numbers, given with the hope of helping a better knowledge of the set of prime numbers. The importance of the remainder function in the computational definitions of arithmetic functions leads to reflections on the importance of fundamental operations - addition and multiplication - of natural numbers and the importance of inverse functions - subtraction and division. In turn, these operations can be seen as functions of two variables on the set of natural numbers. From here, readers are invited to reflect on the problem of the origin of natural numbers, the origin based on revelation or the origin provided by set theory, although this may also be a revelation.

Translated sums of primitive sets

The Erd\H{o}s primitive set conjecture states that the sum $f(A) = \sum_{a\in A}\frac{1}{a\log a}$, ranging over any primitive set $A$ of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and Mechik proved that the translated Erd\H{o}s conjecture for the sum $f(A,h) = \sum_{a\in A}\frac{1}{a(\log a+h)}$ is false starting at $h=81$, by comparison with semiprimes. In this note we prove that such falsehood occurs already at $h= 1.04\cdots$, and show this translate is best possible for semiprimes. We also obtain results for translated sums of $k$-almost primes with larger $k$.

New consequences of prime-counting function

Our objective in this paper is to study a particular set of prime numbers, namely \ left \ {p \ in \ mathbb {P} \ \ text {and} \ \ pi (p) \ notin \ mathbb {P} \ right \} \ !.As a consequence, estimations of the form \ sum {f (p)}with p being prime belonging to this set are derived.

Prime Representing Polynomial

Summary The main purpose of formalization is to prove that the set of prime numbers is diophantine, i.e., is representable by a polynomial formula. We formalize this problem, using the Mizar system [1], [2], in two independent ways, proving the existence of a polynomial without formulating it explicitly as well as with its indication. First, we reuse nearly all the techniques invented to prove the MRDP-theorem [11]. Applying a trick with Mizar schemes that go beyond first-order logic we give a short sophisticated proof for the existence of such a polynomial but without formulating it explicitly. Then we formulate the polynomial proposed in [6] that has 26 variables in the Mizar language as follows (w·z+h+j−q)2+((g·k+g+k)·(h+j)+h−z)2+(2 · k 3·(2·k+2)·(n + 1)2+1−f 2)2+ (p + q + z + 2 · n − e)2 + (e 3 · (e + 2) · (a + 1)2 + 1 − o 2)2 + (x 2 − (a 2 −′ 1) · y 2 − 1)2 + (16 · (a 2 − 1) · r 2 · y 2 · y 2 + 1 − u 2)2 + (((a + u 2 · (u 2 − a))2 − 1) · (n + 4 · d · y)2 + 1 − (x + c · u)2)2 + (m 2 − (a 2 −′ 1) · l 2 − 1)2 + (k + i · (a − 1) − l)2 + (n + l + v − y)2 + (p + l · (a − n − 1) + b · (2 · a · (n + 1) − (n + 1)2 − 1) − m)2 + (q + y · (a − p − 1) + s · (2 · a · (p + 1) − (p + 1)2 − 1) − x)2 + (z + p · l · (a − p) + t · (2 · a · p − p 2 − 1) − p · m)2 and we prove that that for any positive integer k so that k + 1 is prime it is necessary and sufficient that there exist other natural variables a-z for which the polynomial equals zero. 26 variables is not the best known result in relation to the set of prime numbers, since any diophantine equation over ℕ can be reduced to one in 13 unknowns [8] or even less [5], [13]. The best currently known result for all prime numbers, where the polynomial is explicitly constructed is 10 [7] or even 7 in the case of Fermat as well as Mersenne prime number [4]. We are currently focusing our formalization efforts in this direction.