Abstract: 1. Early Times.- 2. Dirichlet's Theorem on Primes in Arithmetic Progressions.- 3. ?ebysev's Theorem.- 4. Riemann's Zeta-function and Dirichlet Series.- 5. The Prime Number Theorem.- 6. The Turn of the Century.- References.- Author Index.

Abstract: We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, lim inf n!1 pnC1 pn logpn D0:

Abstract: We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.

Abstract: After many unsuccessful attempts to obtain a proof in the late 30s-early 40s the following weaker version of The Burnside Problem was studied: Is it true that there are only finitely many 7??-generated finite groups of exponent nl In other words the question is whether there exists a universal finite m-generated group of exponent n having all other finite m-generated groups of exponent n as homomorphic images. Later (thanks to W. Magnus [35]) this question became known as The Restricted Burnside Problem. In 1964 E. S. Golod gave a negative answer to The General Burnside Problem (cf. [9]). Since then a considerable array of infinitely generated periodic groups was constructed by other authors (cf. Alyoshin [2], Suschansky [44], Grigorchuk [ll],Gupta-Sidki [54]). In 1968 P. S. Novikov and S. I. Adian [39] constructed counter-examples to The Burnside Problem for groups of odd exponents n > 4381 (now for odd exponents n > 115, cf. I. Lysenok [33]). Olshansky's Monsters (cf. [40]) shows how wildly periodic groups may behave. At the same time there were two major reasons to believe that The Restricted Burnside Problem would have a positive solution. One of these reasons was the reduction theorem obtained by Ph. Hall and G. Higman [14]. Let n = p\ . ..pf, where /;,• are distinct prime numbers, iq > 1, and assume that (a) The Restricted Burnside Problem for groups of exponents pf has a positive solution, (b) there are only a finite number of finite simple groups of exponent n, (c) the factor group Out(G) = Aut(G)/Inn(G) is solvable for any finite simple group of exponent n. Then The Restricted Burnside Problem for groups of exponent n also has positive solution. Another reason was the close relation of The Problem to Lie algebras. Suppose n = p, where p is a prime number. Then the finite group G of exponent p is clearly nilpotent. It is easy to see that it is sufficient to find an upper bound

Abstract: Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that liminf_{q \to \infty} q . |q|_p . ||q x|| = 0 for all real numbers x. We show that with the additional factor of log q.loglog q the statement is false. Indeed, our main result implies that the set of x for which liminf_{q\to\infty} q . log q . loglog q. |q|_p . ||qx|| > 0 is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.

Abstract: We prove universality for L-functions \( \mathcal{L} \) from the Selberg class satisfying some mild condition on the Dirichlet coefficients (which might be considered as a prime number theorem for \( \mathcal{L} \)). This generalizes a previous universality theorem by the second author, where the L-function was assumed to have a polynomial Euler product satisfying the Ramanujan hypothesis.

Abstract: We aim at using the problems from exact Ramsey theory, concerned with computing Ramsey-type numbers, as a rich source of test problems for SAT solving, targeting especially hard problems. Particularly we consider the links between Ramsey theory in the integers, based on van der Waerden’s theorem, and (boolean, CNF) SAT solving. Based on Green-Tao’s theorem (“the primes contain arbitrarily long arithmetic progressions”) we introduce the Green-Tao numbers grtmk1, ..., km, which in a sense combine the strict structure of van der Waerden problems with the quasi-randomness of the distribution of prime numbers. In general the problem sizes become quickly infeasible here, but we show that for transversal extensions these numbers only grow linearly, thus having a method at hand to produce more problem instances of feasible sizes. Using standard SAT solvers (look-ahead, conflict-driven, and local search) we determine the basic Green-Tao numbers. It turns out that already for this single case of a Ramsey-type problem, when considering the best-performing solvers a wide variety of solver types is covered. This is different to van der Waerden problems, where apparently only simple look-ahead solvers succeed (regarding complete methods). For m>2 the problems are non-boolean, and we introduce the generic translation scheme for translating non-boolean clause-sets to boolean clause-set. This general method offers an infinite variety of translations (“encodings”) and covers the known methods. In most cases the special instance called nested translation proved to be far superior over its competitors (including the direct translation).

Abstract: We introduce the exterior degree of a finite group G to be the probability for two elements g and g′ in G such that g ∧ g′ = 1, and we shall state some results concerning this concept. We show that if G is a non-abelian capable group, then its exterior degree is less than 1/p, where p is the smallest prime number dividing the order of G. Finally, we give some relations between the new concept and commutativity degree, capability, and the Schur multiplier.

Abstract: Let n ≥ 0, let ω be a nonempty set of prime numbers and let τ be a subgroup functor (in Skiba’s sense) such that all subgroups of any finite group G contained in τ (G) are subnormal in G. It is shown that the lattice of all τ-closed n-multiply ω-composite formations is algebraic and modular.

Abstract: In this paper, we continue our study of the pro-$\Sigma$ fundamental groups of configuration spaces associated to a hyperbolic curve, where $\Sigma$ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater--Schneps. More precisely, we show that if one restricts one's attention to outer automorphisms of such a pro-$\Sigma$ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain ``fiber subgroups'' (i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces) as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from $n+1$ to $n \ge 1$ (respectively, $n \ge 2$), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if $n \ge 4$. The key tool in the proof is a combinatorial version of the Grothendieck conjecture proven in an earlier paper by the author, which we apply to construct certain canonical sections.

Abstract: In this article we study left and right 4-Engel elements of a group. In particular, we prove that ⟨a, a b ⟩ is nilpotent of class at most 4, whenever a is of finite order and b ±1 are right 4-Engel elements or a ±1 are left 4-Engel elements and b is an arbitrary element of G. Furthermore, we prove that for any prime p and any element a of finite p-power order in a group G such that a ±1 ∈ L 4(G), a 4, if p = 2, and a p , if p is an odd prime number, is in the Baer radical of G.

Abstract: Let G be a possibly disconnected algebraic group over an algebraically closed field k of characteristic p > 0, such that its neutral connected component, H = G0, is a unipotent group. We recall that an algebraic group over k is defined to be a smooth group scheme of finite type over k. Let us fix a prime number l 6= p. If X is a k-scheme, we use D(X) to denote the bounded derived category of Ql-complexes on X. If the group G acts on X, we use DG(X) to denote G-equivariant bounded derived category of Ql-complexes on X.

Abstract: We find a polynomial in three variables whose values at nonnegative integers satisfy the Erdős-Straus Conjecture. Although the perfect squares are not covered by these values, it allows us to prove that there are arbitrarily long sequence of consecutive numbers satisfying the Erdős-Straus Conjecture. We conjecture that the values of this polynomial include all the prime numbers of the form $4q+5$, which is checked up to $10^{14}$. A greedy-type algorithm to find an Erdős-Straus decomposition is also given; the convergence of this algorithm is proved for a wide class of numbers. Combining this algorithm with the mentioned polynomial we verify that all the natural numbers $n$, $2\le n\le 2\times 10^{14}$, satisfy the Edős-Straus Conjecture.

Abstract: We present low complexity formulae for the computation of cubing and cube root over IF3m constructed using special classes of irreducible trinomials, tetranomials and pentanomials. We show that for all those special classes of polynomials, field cubing and field cube root operation have the same computational complexity when implemented in hardware or software platforms. As one of the main applications of these two field arithmetic operations lies in pairing-based cryptography, we also give in this paper a selection of irreducible polynomials that lead to low cost field cubing and field cube root computations for supersingular elliptic curves defined over IF3m, where m is a prime number in the pairing-based cryptographic range of interest, namely, m ∈ [47, 541].

Abstract: Let be a prime number. We show that a result of Teulie is nearly best possible by constructing a -adic number such that and are uniformly simultaneously very well approximable by rational numbers with the same denominator. The same conclusion was previously reached by Zelo in his PhD thesis, but our approach using -adic continued fractions is more direct and simpler.

Abstract: One of the main tasks in the analysis of prime numbers distribution is to single out hidden rules and regular features like periodicity, typical patterns, trends, etc. The existence of fractal shapes, patterns and symmetries in prime numbers distribution are discussed.

Abstract: In this article, we prove that in content extentions minimal primes extend to minimal primes and discuss zero-divisors of a content algebra over a ring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also examine the preservation of diameter of zero-divisor graph under content extensions.

Abstract: A classical conjecture in the representation theory of finite groups, the McKay conjecture, states that for any finite group and prime number p the number of complex irreducible characters of degree prime to p is equal to the number of complex irreducible characters of degree prime to p of the normalizer of a p-Sylow subgroup. Recently a reduction theorem was proved by Isaacs, Malle and Navarro: If all simple groups are “good”, then the McKay conjecture holds. In this work we are concerned with the problem of goodness for finite groups of Lie type in their defining characteristic. A simple group is called “good” if certain equivariant bijections between the involved character sets exist. We present a structural approach to the construction of such a bijection by utilizing the so-called “Steinberg-Map”. This yields very natural bijections and we prove most of the desired properties.

Abstract: This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. Second, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. The first algorithm extends the recent work by Koutis (2008) and Williams (2009) on testing multilinear monomials. Group algebra is exploited in the algorithm designs, in corporation with the randomized polynomial identity testing over a finite field by Agrawal and Biswas (2003), the deterministic noncommunicative polynomial identity testing by Raz and Shpilka (2005) and the perfect hashing functions by Chen {\em at el.} (2007). Finally, we prove that testing some special types of multilinear monomial is W[1]-hard, giving evidence that testing for specific monomials is not fixed-parameter tractable.

Abstract: Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and p - α divides N - α for each prime divisor p of N. We are concerned, here, with both a numerical and theoretical study of composite squarefree Korselt numbers. The paper contains two main results. The first one shows that for α ∈ ℤ\{0}, the following properties hold: (i) If α ≤ 1, then each composite squarefree Kα-number has at least three prime factors. (ii) Suppose that α > 1. Let p < q be two prime numbers and N ≔ pq. If N is an α-Korselt number, then p < q ≤ 4α - 3. In particular, there are only finitely many α-Korselt numbers with exactly two prime factors. Let α ∈ ℕ\{0}; by an α-Williams number (Wα-number, for short) we mean a positive integer which is both a Kα-number and a K-α-number. Our second main result shows that if p, 3p - 2, 3p + 2 are all prime, then their product is a (3p)-Williams number.

Abstract: We consider the links between Ramsey theory in the integers, based on van der Waerden's theorem, and (boolean, CNF) SAT solving. We aim at using the problems from exact Ramsey theory, concerned with computing Ramsey-type numbers, as a rich source of test problems, where especially methods for solving hard problems can be developed. In order to control the growth of the problem instances, we introduce "transversal extensions" as a natural way of constructing mixed parameter tuples (k_1, ..., k_m) for van-der-Waerden-like numbers N(k_1, ..., k_m), such that the growth of these numbers is guaranteed to be linear. Based on Green-Tao's theorem we introduce the "Green-Tao numbers" grt(k_1, ..., k_m), which in a sense combine the strict structure of van der Waerden problems with the (pseudo-)randomness of the distribution of prime numbers. Using standard SAT solvers (look-ahead, conflict-driven, and local search) we determine the basic values. It turns out that already for this single form of Ramsey-type problems, when considering the best-performing solvers a wide variety of solver types is covered. For m > 2 the problems are non-boolean, and we introduce the "generic translation scheme", which offers an infinite variety of translations ("encodings") and covers the known methods. In most cases the special instance called "nested translation" proved to be far superior.

Abstract: In this paper we examine the connectedness of arithmetic progressions in the following topologies: Furstenberg's topology on the set of integers, Golomb's topology D on the set of positive integers, and Kirch's topology Don the set of positive integers. Immediate consequences of these studies are theorems concerning the connectedness and the locally connectedness of the topologies D and Dproved by S. Golomb in 1959 and A. M. Kirch in 1969. 1. Preliminaries The letters Z, N and N0 denote the sets of integers, positive integers, and non-negative integers, respectively. The symbol �(a) denotes the set of all prime factors of a ∈ N. For all a,b ∈ N, we use the symbols (a,b) and lcm(a,b) to denote the greatest common divisor of a and b and the least common multiple of a and b, respectively. Moreover, for all a,b ∈ N, the symbols {an + b} and {an} stand for the infinite arithmetic progressions:

Abstract: This paper formalizes the optimal base problem, presents an algorithm to solve it, and describes its application to the encoding of Pseudo-Boolean constraints to SAT. We demonstrate the impact of integrating our algorithm within the Pseudo-Boolean constraint solver MINISAT+. Experimentation indicates that our algorithm scales to bases involving numbers up to 1,000,000, improving on the restriction in MINISAT+ to prime numbers up to 17. We show that, while for many examples primes up to 17 do suffice, encoding with respect to optimal bases reduces the CNF sizes and improves the subsequent SAT solving time for many examples.

Abstract: Let $p$ be an odd prime number. We prove that for $m\equiv1\mod p$, $x^m$ is perfectly nonlinear over $\mathbb{F}_{p^n}$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\in\mathbb{N}$. First, we study singularities of $f(x,y)=\frac{(x+1)^m-x^m-(y+1)^m+y^m}{x-y}$ and we use Bezout theorem to show that for $m eq 1+p^l$, $f(x,y)$ has an absolutely irreducible factor. Then by Weil theorem, f(x,y) has rationnal points such that $x eq y$ which means that $x^m$ is not PN.

Abstract: This thesis deals with the classical problem of prime numbers represented by polynomials. It consists of three parts. In the first part I collected many results about the problem. Some of them are quite recent and this part can be considered as a survey of the state of the art of the subject. In the second part I present two results due to P. Pleasants about the cubic polynomials with integer coefficients in several variables. The aim of this part is to simplify the works of Pleasants and modernize the notation employed. In such a way these important theorems are now in a more readable form. In the third part I present some original results related with some algebraic invariants which are the key-tools in the works of Pleasants. The hidden diophantine nature of these invariants makes them very difficult to study. Anyway some results are proved. These results make the results of Pleasants somewhat more effective.