Random differences in Szemerédi’s theorem and related results

The vital characteristic of randomness is unpredictability. Thus any regularity will compromise the application of random numbers. Quantum random number generators (QRNGs) can provide intrinsic unpredictable randomness based on the nature of quantum physics, while pseudo-random number generator can not due to the origination of deterministic algorithms or physical processes. However, commonly used traditional test suits are rigorously to test the statistical properties, and the unpredictability of random numbers is still difficult to test. To verify which sources of random numbers are truly unpredictable, in this paper, we propose a new randomness testing method with the artificial neural network (ANN). Random number sequences generated by four different kinds of sources are tested, which are the natural number λ, the linear congruence generator (LCG) pseudo-random algorithm, the Mersenne Twister (MT) pseudorandom algorithm and the QRNG based on vacuum noise, respectively. The testing results indicate that the random sequences from natural number λ and LCG fail to pass our randomness test, while the other two kinds of sequences pass the test successfully due to the relatively simple structure of our ANN. As the complexity of the ANN and the amount of computing power involved increasing, this test method has potential to predict the MT random numbers, and it is also expected that the quantum random numbers can not pass the test with unpredictability no matter how complex the ANN is.

This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2 k + p, where k is a positive integer and p is an odd prime. Erdős ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2 k +p, and give a quantitative form of Romanoff’s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdős problem is affirmative.

Si q≥2 est un nombre entier, on designe par Sq(n) la somme des chiffres en base q du nombre entier naturel n et par vq(n) sa valuation q-adique. L’objectif de cet article est d’etudier des sommes d’exponentielles de la forme ∑n≤xexp(2iπ(lmSq(n)+km′Sq(n+1)+θn)) afin d’en deduire certaines proprietes statistiques des nombres entiers n pour lesquels Sq(n) et Sq(n+1) appartiennent a des progressions arithmetiques donnees. Ceci permet d’etendre les resultats obtenus par Gelfond en 1968 et ceux obtenus par Mauduit-Sarkozy en 1996.

Erdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

Introduction 1. Primes in arithmetic progressions 2. The main theorems II.Oscillation theorems for sifting functions 3. The statements 4. Proof of Theorem B2 5. Proof of Proposition 3.1 6. Proof of Proposition 3.2 7. Proof of Theorem C: preliminaries 8. Proof of Theorem C: completion 9. Proof of Theorem Bl 10.Proof of Theorem B3 III.Oscillation theorems for primes 11.Proof of Theorem Al 12. Proof of Theorem A2 13.Proofs of Theorem A3 and Proposition 2.1 I. INTRODUCTION 1. PRIMES IN ARITHMETIC PROGRESSIONSFor x ~ 2 real, q a positive integer, and a an integer coprime to q, let 0 and ~ be defined by (1.1) O(x; q, a) = E logp = qJZq) (1 +~(x; q, a)), p~x p=amodq

On square-free arithmetic progressions in infinite words

In their valuable book Board games around the world . Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia . One such investigation gives rise to the following problem: Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression). The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.

Strongly sufficient sets and the distribution of arithmetic sequences in the [formula omitted] graph

This thesis consists of three topics. The first one is on quadratic Waring-Goldbach problems. The second topic is about some additive problems involving fourth powers. The last topic is to consider an average result for the divisor problem in arithmetic progressions.\n\n\n\nChapter 1 is an introduction. In Chapter 2 the Lagrange equation with almost prime variables is studied, based on a method of Heath-Brown and Tolev. The techniques involve the circle method and the sieve method in analytic number theory. It is established that every sufficiently large integer, congruent to 4 modulo 24, can be represented as the sum of four squares of integers, each of which has at most four prime factors. In order to derive the desired result, it is important to study the asymptotic formula for a smooth counting function involving the sum of one square of a prime number and three squares of integers in arithmetic progressions. For this purpose, the square sieve of Heath-Brown and the Kloosterman refinement are employed. This is the main task of Chapter 3.\n\n\n\nChapter 4 is devoted to the investigation on the sum of four squares of primes and K powers of two. Inspired by the work of Wooley, this problem is dealt with by applying the linear sieve instead of the four dimensional vector sieve.\n\n\n\nThe theme of Chapter 5 is the three squares theorem with almost prime variables, firstly investigated by Blomer and Br?udern. A sharper result is obtained, in this chapter, by developing the three dimensional weighted sieve.\n\n\n\nIn Chapter 6, the main concern is the exceptional set for the sum of fourth powers. By developing Vaughan's p-adic iteration method, some new estimations are established. The new estimations improve upon the earlier result obtained by Kawada and Wooley. As a consequence of the new estimates, it is established that every sufficiently large integer under a natural congruence condition can be expressed as a sum of six fourth powers of primes and six fourth powers of integers. Moreover, a pair of diagonal quartic forms is considered. In Chapter 7, it is shown that a pair of diagonal quartic equations with 23 variables has nonzero integral solutions under certain reasonable conditions.\n\n\n\nThe purpose of Chapter 8 is to consider a Waring-Goldbach type problem involving twelve fourth powers of primes and one fourth power of a positive integer. The idea of proof is to combine Chen's switching principle and Iwaniec's linear sieve. A new result is obtained, which refines that of Ren and Tsang on this topic.\n\n\n\nIn the last chapter, a distribution problem for the divisor function on arithmetic progressions is investigated. Denote by ?q,b(X) the error term in the divisor problem in the arithmetic progression b(mod q). New asymptotic formulae for the variance A(q, X) =?qb=1|?q,b(x)|2 for X1=4+E ?· q?·X1=2-E and X1=2+E ? q ·?X1-E are derived. The distinct behaviors of A(q, X) in these two ranges are unveiled.

In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in a fixed arithmetic progression. Contrary to the classical setting of reductions of one-parameter families over the rationals, where it is conjectured by Steven J. Miller that the bias is always negative, we prove that in our setting the bias is positive for a positive density of arithmetic progressions and negative for a positive density of arithmetic progressions. Along the way, we obtain explicit formulas for moments of traces of Frobenius of elliptic curves over finite fields in arithmetic progressions and related moments of Hurwitz class numbers in arithmetic progressions, the distribution of which are of independent interest.

Szemerédi's famous theorem on arithmetic progressions asserts that every subset of integers of positive asymptotic density contains arithmetic progressions of arbitrary length. His remarkable theorem has been developed into a major new area of combinatorial number theory. This is the topic of the present survey.

A new methodology has been recently proposed for the efficient generation of multiple pseudo-random bit sequences that are statistically uncorrelated [R. Kuehnel, March 2003]. Random sequences that are uniformly distributed over a range [0,K], where K+ 1 is a power of 2, can be constructed by forming a vector of M independent bit sequences, where M = log/sub 2/(K + 1). In this paper we demonstrate that this method of construction represents a special case of a more generalized approach in which K can be any positive integer. The procedures described here can be used to efficiently generate multiple independent random sequences that are uniformly distributed over any range.

Ternary quadratic forms representing a given arithmetic progression

anXrn(ϊi> ''' >7n e E and au '-,aneQ will be called a central .^-polynomial, and the set of all such functions will be denoted Z^~E. A set EaΓ is a central Sidon set when the norms || H*, and || |U(||/|U — ΣlαJ> where /— Σ^nXrJ are equivalent on z^Ei and it is a central Λ(p) set when the norms || ^ and || \\p are equivalent on Z ^Έ* When G is abelian the algebraic structure of Λ(p) and Sidon set has been studied extensively. In this paper the structure of central Λ(p) sets is investigated in terms of the hypergroup structure of Γ. In particular it is shown that central Λ(p)(p > 2) sets cannot contain arbitrarily long arithmetic progressions. 1* Preliminary remarks* Following Helgason [2] we shall say that a set S is hypergroup if to any pair (a, β) of elements from S there corresponds a measure μa,β on S. For Γ, a hypergroup structure is induced by the decomposition of tensor products. Thus if a, β e Γa® β — φ re r[τ: <2(x) β]j, where [y:a®β] is a nonnegative integer which is called the multiplicity of 7 in a (x) β, and the measure assigned to the pair (a, β) is the discrete measure whose mass at y is [j: a (x) β]. From the elementary properties of characters we write XTl Xϊn = Xnto—torn ~ Σrer [T: ΎX (x) (x) ΊnXXγ- We shall denote by 1 the class of the trivial one dimensional representation, and by 7 the class containing the conjugates of representatio ns in j. All the basic facts about representations needed in this paper may be found in [3].