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For nonnegative integers h , g with h ≥ 2 , a set A of nonnegative integers is defined as a B h [ g ] sequence if, for every nonnegative integer n , the number of representations of n with the form n = a 1 + a 2 + ⋯ + a h is no larger than g , where a 1 ≤ ⋯ ≤ a h and a i ∈ A for i = 1 , 2 , ⋯ , h. Let Z be the set of integers and N be the set of positive integers. In 2013, by introducing the method of Inserting Zeros Transformation , Cilleruelo and Nathanson obtained the following result: let f : Z → N ⋃ { 0 , ∞ } be any function such that lim inf | n | → ∞ f (n) ≥ g and let B be any B h [ g ] sequence. Then, for any decreasing function ϵ (x) → 0 as x → ∞ , there exists a sequence A of integers such that r A , h (n) = f (n) for all n ∈ Z and A (x) ≫ B (x ϵ (x)). Recently, Nathanson further considered Sidon sets for linear forms. In this paper, we apply the Inserting Zeros Transformation into Sidon sets for linear forms and generalize the above result related to the inverse problem of representation functions. For a video summary of this paper, please visit https://youtu.be/KRtew2rLbXY. [ABSTRACT FROM AUTHOR]

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ , where μ ≥ C log ⁡ λ , such that intervals [ λ , λ + μ ] do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in R 2 that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in R 2. Specifically, we establish the existence of annuli { x ∈ R 2 : λ ≤ | x | 2 ≤ λ + κ } with arbitrarily large λ and κ ≥ C λ s for 0 < s < 1 4 , satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold s = 1 4. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in R 3. [ABSTRACT FROM AUTHOR]

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset S of Z. Bhargava's factorials k ! S are invariants, constructed using the notion of p -orderings of S where p is a prime. This paper defines b -orderings of any nonempty subset S of Z for all integers b ≥ 2 , as well as "extreme" cases b = 1 and b = 0. It defines generalized factorials k ! S , T and generalized binomial coefficients ( k + ℓ k ) S , T as nonnegative integers, for all nonempty S and allowing only b in T ⊆ N. It computes b -ordering invariants when S is Z and when S is the set of all primes. [ABSTRACT FROM AUTHOR]

Suppose that p is an odd prime, and R = G R (p 2 , p 2 r) is a Galois ring of characteristic p 2 , whose cardinality is p 2 r. For any subset D ⊆ R , let N (D , n , b) and N f (D , n , b) , respectively, be the number of n -subsets, denoted by S , in D such that ∑ x ∈ S x = b and ∑ x ∈ S f (x) = b , where f (x) ∈ R [ x ] is a polynomial with degree d. In this paper, we give the asymptotic formulae of N (D , n , b) and N f (D , n , b) under some certain constraints of D and f (x). The special cases D = { a k : a ∈ R ⁎ } and f (x) = x k were studied by the authors in a previous paper, where R ⁎ is the subset of all invertible elements of R and k is a given positive integer, which extends the former one in a more general setting. These extensions benefit from a refined form of the Li–Wan new sieve as well as a variant of the Weil bound on Galois rings. [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

Let k , t be coprime integers, and let 1 ≤ r ≤ t. We let D k × (r , t ; n) denote the total number of parts among all k -indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3] , an asymptotic estimate for D k × (r , t ; n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L -functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k -indivisible partitions of n as n becomes large. [ABSTRACT FROM AUTHOR]

Let b ≥ 2 be an integer and A = (a n) n = 1 ∞ be a strictly increasing subsequence of positive integers with η : = lim sup n → ∞ a n + 1 a n < + ∞. For each irrational real number ξ , we denote by v ˆ b , A (ξ) the supremum of the real numbers v ˆ for which, for every sufficiently large integer N , the equation ‖ b a n ξ ‖ < (b a N ) − v ˆ has a solution n with 1 ≤ n ≤ N. For every v ˆ ∈ [ 0 , η ] , let V ˆ b , A (v ˆ) ( V ˆ b , A ⁎ (v ˆ)) be the set of all real numbers ξ such that v ˆ b , A (ξ) ≥ v ˆ ( v ˆ b , A (ξ) = v ˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ). When η = 1 , we prove that the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) are equal to (1 − v ˆ 1 + v ˆ ) 2 for any v ˆ ∈ [ 0 , 1 ]. When η > 1 and lim n → ∞ ⁡ a n + 1 a n exists, we show that the Hausdorfff dimension of V ˆ b , A (v ˆ) is strictly less than (η − v ˆ η + v ˆ ) 2 for some v ˆ , which is different with the case η = 1 , and we give a lower bound of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) for any v ˆ ∈ [ 0 , η ]. Furthermore, we show that this lower bound can be reached for some v ˆ. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p -adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m ∈ N such that p ℓ α is an infinite loop mod p m , for all ℓ ∈ N ∪ { 0 }. [ABSTRACT FROM AUTHOR]

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete "stick breaking") subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior. [ABSTRACT FROM AUTHOR]

Let D , k be integers with D square free and k even. Let N be a positive integer so that (N , D) = 1 when D has residue one modulo four and (N , 4 D) = 1 when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum S X (α , β ; π) attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by D is computed. First a Voronoi summation formula is derived that expresses S X (α , β ; π) in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of S X (α , β ; π) as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N , which is a special case of the multiplicity one theorem. [ABSTRACT FROM AUTHOR]

Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

In a paper from twenty years ago, Brown and Myers proved the clean and uniform result that if m > 1 is an integer, then every curve of the form y 2 = x 3 − x + m 2 has Mordell-Weil rank at least 2. In this article, we further our investigation into the more general family of curves, in which the polynomial x 3 − x is replaced by an arbitrary cubic which is squarefree and has three integral roots. In particular, a new strategy will unveil a subfamily of curves with surprisingly large average rank, and one startling example. [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group. Let F (G) denote the monoid of all sequences over G. For Ω ⊂ F (G) , let d Ω (G) be the smallest integer t such that every sequence S over G of length | S | ≥ t has a subsequence in Ω. A sequence S over G is a weak-regular sequence if v g (S) ≤ ord (g) for every g ∈ G. Let N (G) be the smallest integer t such that every weak-regular sequence S over G of length | S | ≥ t has a nontrivial zero-sum subsequence T with v g (T) = v g (S) > 0 for some g | S. The invariant N (G) was formulated in a recent paper [4] to study d Ω (G) and has its own interest, and N (G) has been determined for the cyclic groups of prime order there. In this paper, among other results, we shall determine the precise value of N (G) for cyclic groups of prime power order and for elementary abelian groups of rank two. [ABSTRACT FROM AUTHOR]

• Integer lattice. • Discrepancy. • Annuli. • Gauss circle problem. Given a domain Ω ⊆ R 2 , let D (Ω , x , R) be the number of lattice points from Z 2 in R Ω − x , for R ≥ 1 and x : = (x 1 , x 2) ∈ T 2 , minus the area of R Ω: D (Ω , x , R) = # { (j , k) ∈ Z 2 : (j − x 1 , k − x 2) ∈ R Ω } − R 2 | Ω |. We call ∫ T 2 | D (Ω , x , R) | p d x the p -th moment of the discrepancy function D. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of D is bounded by O (R 2 log ⁡ R) , and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions. In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p -th moments of lattice point discrepancy of annuli of radius R , and any fixed thickness 0 < t < 1 for p ≥ 2. For a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q. [ABSTRACT FROM AUTHOR]

In all available papers, on power integral bases of any pure sextic number fields K generated by a complex root α of a monic irreducible polynomial f (x) = x 6 − m ∈ Z [ x ] , it was assumed that the rational integer m ≠ ∓ 1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free rational integer is omitted. We start by calculating an integral basis of Z K ; the ring of integers of K. In particular, we characterize when Z K = Z [ α ] , that is when Z K is monogenic and generated by α. We give sufficient conditions on m , which warranty that K is not monogenic. We finish the paper by investigating the case, where m = e 5 and e ≠ ∓ 1 is a square free rational integer. [ABSTRACT FROM AUTHOR]

This paper elaborates on a sieving technique that has first been applied in 2018 for improving bounds on deterministic integer factorization. We will generalize the sieve in order to obtain a polynomial-time reduction from integer factorization to a specific instance of the multiple choice subset-sum problem. As an application, we will improve upon special purpose factorization algorithms for integers composed of divisors with small difference. In particular, we will refine the runtime complexity of Fermat's factorization algorithm by a large subexponential factor. Our first procedure is deterministic, rigorous, easy to implement and has negligible space complexity. Our second procedure is heuristically faster than the first, but has non-negligible space complexity. [ABSTRACT FROM AUTHOR]

A generalization of the well–known Fibonacci sequence is the ℓ -Fibonacci sequence (F m (ℓ)) m whose first ℓ terms are 0 , ... , 0 , 1 and each term afterwards is the sum of the preceding ℓ terms. The k -Pell sequence (P n (k)) n , which is a generalization of the classical Pell sequence, can be defined similarly. In this paper, we find all coincidences between these two families of sequences. That is, we find all the solutions of the Diophantine equation P n (k) = F m (ℓ) in positive integers n , k , m , ℓ with k , ℓ ≥ 2. This paper continues and extends prior results which dealt with the above problem for some particular cases of k and ℓ. In particular, it extends the previous work [2] that found all Fibonacci numbers in the Pell sequence. [ABSTRACT FROM AUTHOR]

By using the work of Frantzikinakis and Wierdl, we can see that for all d ∈ N , α ∈ (d , d + 1) , and integers k ≥ d + 2 and r ≥ 1 , there exist infinitely many n ∈ N such that the sequence (⌊ (n + r j) α ⌋) j = 0 k − 1 is represented as ⌊ (n + r j) α ⌋ = p (j) , j = 0 , 1 , ... , k − 1 , by using some polynomial p (x) ∈ Q [ x ] of degree at most d. In particular, the above sequence is an arithmetic progression when d = 1. In this paper, we show the asymptotic density of such numbers n as above. When d = 1 , the asymptotic density is equal to 1 / (k − 1). Although the common difference r is arbitrarily fixed in the above result, we also examine the case when r is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate asymptotically the sum ∑ d ≤ X (∑ n ≤ Y μ (n) (8 d n)) 2 , where (8 d n) is the Kronecker symbol and d runs over positive, odd, square-free integers. [ABSTRACT FROM AUTHOR]

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by x over which there exist elliptic curves with good reduction everywhere and rational j -invariant is ≫ x log − 1 2 ⁡ (x). In this paper, we assume the abc -conjecture to show the sharp asymptotic ∼ c x log − 1 2 ⁡ (x) for this number, obtaining formulae for c in both the real and imaginary cases. Our method has three ingredients: (1) We make progress towards a conjecture of Granville: Given a fixed elliptic curve E / Q with short Weierstrass equation y 2 = f (x) for reducible f ∈ Z [ x ] , we show that the number of integers d , | d | ≤ D , for which the quadratic twist d y 2 = f (x) has an integral non-2-torsion point is at most D 2 3 + o (1) , assuming the abc -conjecture. (2) We apply the Selberg–Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes. (3) We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most x is O (x 1 − ϵ) for some ϵ > 0. We also exhibit a matching lower bound. If instead of the abc -conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient c above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias. [ABSTRACT FROM AUTHOR]

Let 1 ⩽ a < b be two relatively prime integers. Sylvester found that a b − a − b is the largest integer which can not be represented by a x + b y (x , y ∈ Z ⩾ 0) about 160 years ago and this number shall be denoted by g a , b. Let N (a , b) = { n : n ⩽ g a , b , n = a x + b y , x , y ∈ Z ⩾ 0 } and π a , b be the number of primes in N (a , b). Recently, Ramírez Alfonsín and Skałba proved that π a , b ≫ ε g a , b (log ⁡ g a , b) 2 + ε for any fixed ε > 0. They further conjectured that the order of the magnitude of π a , b is 1 2 π (g a , b) , where π (x) is the number of all primes up to x. In this paper, we show that the conjecture is true for almost all pairs a , b with 1 ⩽ a < b and (a , b) = 1. The proofs rely heavily on the Bombieri–Vinogradov theorem and Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

For a number field F and an odd prime number p , let F ˜ be the compositum of all Z p -extensions of F and Λ ˜ the associated Iwasawa algebra. Let G S (F ˜) be the Galois group over F ˜ of the maximal extension which is unramified outside p -adic and infinite places. In this paper we study the Λ ˜ -module X S (− i) (F ˜) : = H 1 (G S (F ˜) , Z p (− i)) and its relationship with X (F ˜ (μ p)) (i − 1) Δ , the Δ : = Gal (F ˜ (μ p) / F ˜) -invariant of the Galois group over F ˜ (μ p) of the maximal abelian unramified pro- p -extension of F ˜ (μ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ ˜ -module X (F ˜ (μ p)) (i − 1) Δ is implied by the existence of a Z p d -extension L with X S (− i) (L) : = H 1 (G S (L) , Z p (− i)) being without torsion over the Iwasawa algebra associated to L , and which contains a Z p -extension F ∞ satisfying H 2 (G S (F ∞) , Q p / Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [ F (μ p) : F ]. This existence is fulfilled for (p , i) -regular fields. [ABSTRACT FROM AUTHOR]

In this paper, we investigate exceptional sets in the biquadratic Waring-Goldbach problem. For example, we prove that all but O (N 60 / 61) integers subject to the necessary local conditions can be represented as the sum of seven biquadrates of primes. Furthermore, the method in this paper can be easily used for exceptional sets for sums of fifth and higher of powers of primes. For a video summary of this paper, please visit https://youtu.be/PGa8P-Px8bk. [ABSTRACT FROM AUTHOR]

For a positive integer k , let f (k) denote the largest integer t such that for every finite abelian group G and every zero-sum free subset S of G , if | S | = k then | Σ (S) | ≥ t. In this paper, we prove that f (k) ≥ 1 6 k 2 , which significantly improves a result of J.E. Olson. We also supply some interesting results on f (k). For a video summary of this paper, please visit https://youtu.be/ZEHZFRUVJKY. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2 , let { F n (k) } n ≥ 2 − k be the k –generalized Fibonacci sequence which starts with 0 , ... , 0 , 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d ≥ 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − d y 2 = ± 1 , which is a k –generalized Fibonacci number, with a couple of parametric exceptions which we completely characterize. This paper extends previous work from [18] for the case k = 2 and [17] for the case k = 3. [ABSTRACT FROM AUTHOR]

For an arbitrary real number α and a positive integer n , the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined by L n (α) (x) = (− 1) n ∑ j = 0 n (n + α) (n − 1 + α) ⋯ (j + 1 + α) (n − j) ! j ! (− x) j. Following the work of Banerjee, Filaseta, Finch and Leidy [2] which described the set A 0 ∪ A ∞ of integer pairs (n , α) for which the discriminant of L n (α) (x) is a nonzero square, where A 0 is finite and A ∞ is explicitly given infinite set, it was conjectured by Banerjee in [1] that for α ≠ − 1 , the only pair (n , α) ∈ A ∞ for which the associated Galois group of L n (α) (x) is not A n is (4 , 23). In this paper, we verify this conjecture for (α , n) ∈ A ∞ with α ∈ { − 2 n , − 2 n − 2 , − 2 n − 4 }. In fact, we prove more general results concerning the irreducibility and Galois groups of the Generalized Laguerre polynomials L n (α) (x) for n ∈ N and integers α such that α ∈ [ − 2 n − 4 , − 2 n ]. The case α = − 2 n − 1 corresponds to the Bessel polynomials which have been studied earlier by Filaseta and Trifonov [7] and Grosswald [9]. Our ideas give a simpler proof of their results. [ABSTRACT FROM AUTHOR]

For any positive integer m , let Z m be the set of residue classes modulo m. For A ⊆ Z m and n ‾ ∈ Z m , let the representation function R A (n ‾) denote the number of solutions of the equation n ‾ = a ‾ + a ′ ‾ with unordered pairs (a ‾ , a ′ ‾) ∈ A × A. Let M be an odd integer with M ≥ 3 and m = 2 M. In this paper, we determine the structure of A , B ⊆ Z m with A ∪ B = Z m and | A ∩ B | = 2 such that R A (n ‾) = R B (n ‾) for all n ‾ ∈ Z m. [ABSTRACT FROM AUTHOR]

For any k ≥ 2 and fixed b = (b 1 , ⋯ , b k) ∈ N k , this paper concerns the number of integer lattice points in Z k which are b -visible from a set of N watch-points. Moreover, for N = 2 and b ˜ = ( b ˜ 1 , ⋯ , b ˜ k) ∈ N k with b ˜ ≠ b , we consider the mixed visibility, that is, the number of integer lattice points which are b -visible and b ˜ -visible from two watch-points respectively. We prove asymptotic formulas for the number of corresponding visible lattice points. Part of our results improves a classical result of Rearick. [ABSTRACT FROM AUTHOR]

Let (F n) n ≥ 0 be the Fibonacci sequence given by F 0 = 0 , F 1 = 1 and F n + 2 = F n + 1 + F n , for all n ≥ 0. In this paper, we find all positive integer solutions (m , n , a , k) of the Diophantine equation F n ± a (10 m − 1) 9 = k ! with 1 ≤ a ≤ 9. Our proof requires lower bounds for nonzero linear forms in two logarithms of algebraic numbers both in the complex and p -adic cases and some computer calculations. [ABSTRACT FROM AUTHOR]

Square-free values of polynomials had been studied by various authors, including Estermann, Heath-Brown and Hooley. For 1 ≤ x , y ≤ H , Tolev proved that the number of the square-free values attained by the polynomial x 2 + y 2 + 1 has the asymptotic formula c 1 H 2 + O (H 4 / 3 + ε) , where is c 1 is an absolute constant and ε is an arbitrary small positive number. The key ingredient of his proof which leads to the elaborate error term is the estimate for the Kloosterman sum. In this paper, by using Tolev's method and some estimate for the Salié sum, we show that for any fixed integer k , there is an absolute constant c 2 such that the number of square-free values of the polynomial x 2 + y 2 + z 2 + k with 1 ≤ x , y , z ≤ H is c 2 H 3 + O (H 7 / 3 + ε). [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are (2 , 8) and (3 , 9). For triples, we prove the following theorem: If a ∉ { 2 , 8 } is a fixed integer larger than 1, then there are only finitely many triples (a , b , c) of pairwise distinct integers larger than 1 such that (a , b , c) , (a + 1 , b + 1 , c + 1) and (a + 2 , b + 2 , c + 2) are each multiplicatively dependent. Moreover, these triples can be determined effectively. [ABSTRACT FROM AUTHOR]

In this paper we study how often the divisor function lies in a set 〈 P 〉 generated by a subset P of the primes by multiplication. We consider a variety of different sets P using results on the average behaviour of multiplicative functions by Wirsing and Delange. In particular, we find out how often d (n) is a power of a given integer. The methods apply more generally to non-negative integer valued multiplicative functions. [ABSTRACT FROM AUTHOR]

Let λ f (n) denote the n th normalized Fourier coefficient of the primitive holomorphic cusp forms of even integral weight k ≥ 2 for the full modular group S L (2 , Z). For x ≥ 1 , we are interested in the sums ∑ n ≤ x λ f (n) ℓ and ∑ a 2 + b 2 ≤ x λ f (a 2 + b 2) ℓ. In this paper, we are able to establish the asymptotic formulae for general cases of the power sum for every positive integer ℓ ∈ N. The special cases of our results improve previous results. [ABSTRACT FROM AUTHOR]

In 1988, Erdős, Nathanson and Sárközy proved that, if A is a set of nonnegative integers with the lower asymptotic density 1 / k , where k is a positive integer, then (k + 1) A must contain an infinite arithmetic progression with difference at most k 2 − k , where (k + 1) A is the set of all sums of k + 1 elements of A , and asked if (k + 1) A must contain an infinite arithmetic progression with difference at most O (k). In this paper, we solve this problem by proving that, for every sufficiently large integer k , there exists a set A of nonnegative integers with the lower asymptotic density 1 / k such that (k + 1) A does not contain an infinite arithmetic progression with difference less than k 1.5. We also study the similar problem when A has the lower asymptotic density 0 < w < 1. For a video summary of this paper, please visit https://youtu.be/B1QJc2IFSIw. [ABSTRACT FROM AUTHOR]

Abstract Let n be a non-negative integer and put p n (x) = ∏ i = 0 n (x + i). In the first part of the paper, for given n , we study the existence of integer solutions of the Diophantine equation y m = p n (x) + ∑ i = 1 k p a i (x) , where m ∈ N ≥ 2 and a 1 < a 2 < ... < a k < n. This equation can be considered as a generalization of the Erdős–Selfridge Diophantine equation y m = p n (x). We present some general finiteness results concerning the integer solutions of the above equation. In particular, if n ≥ 2 with a 1 ≥ 2 , then our equation has only finitely many solutions in integers. In the second part of the paper we study the equation y m = ∑ i = 1 k p a i (x i) , for m = 2 , 3 , which can be seen as an additive version of the equation considered by Erdős and Graham. In particular, we prove that if m = 2 , a 1 = 1 or m = 3 , a 2 = 2 , then for each k − 1 tuple of positive integers (a 2 , ... , a k) there are infinitely many solutions in integers. [ABSTRACT FROM AUTHOR]

Let A be a set of finite integers, define A + A = { a 1 + a 2 : a 1 , a 2 ∈ A } , A − A = { a 1 − a 2 : a 1 , a 2 ∈ A } , and for non-negative integers s and d define s A − d A = A + ⋯ + A ︸ s − A − ⋯ − A ︸ d. A More Sums than Differences (MSTD) set is an A where | A + A | > | A − A |. It was initially thought that the percentage of subsets of [ 0 , n ] that are MSTD would go to zero as n approaches infinity as addition is commutative and subtraction is not. However, in a surprising 2006 result, Martin and O'Bryant proved that a positive percentage of sets are MSTD, although this percentage is extremely small, about 10 − 4 percent. This result was extended by Iyer, Lazarev, Miller, and Zhang [ILMZ] who showed that a positive percentage of sets are generalized MSTD sets, sets for { s 1 , d 1 } ≠ { s 2 , d 2 } and s 1 + d 1 = s 2 + d 2 with | s 1 A − d 1 A | > | s 2 A − d 2 A | , and that in d -dimensions, a positive percentage of sets are MSTD. For many such results, establishing explicit MSTD sets in 1-dimensions relies on the specific choice of the elements on the left and right fringes of the set to force certain differences to be missed while desired sums are attained. In higher dimensions, the geometry forces a more careful assessment of what elements have the same behavior as 1-dimensional fringe elements. We study fringes in d -dimensions and use these to create new explicit constructions. We prove the existence of generalized MSTD sets in d -dimensions and the existence of k -generational sets, which are sets where | c A + c A | > | c A − c A | for all 1 ≤ c ≤ k. We then prove that under certain conditions, there are no sets with | k A + k A | > | k A − k A | for all k ∈ N. For a video summary of this paper, please visit https://youtu.be/rojbhVqN90Q. [ABSTRACT FROM AUTHOR]

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p (n) , the number of partitions of n. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if d is sufficiently large square-free positive rational integer, then there is no positive definite universal quadratic O F -lattice of rank 7 where F = Q (d). [ABSTRACT FROM AUTHOR]

Text For each positive integer n , let L ( n ) be the length of longest arithmetic progressions in the least positive reduced residue system modulo n . In this paper, we give an asymptotic formula for ∑ n ≤ x L ( n ) . This solves a problem of Pongsriiam (2018) [4] . Video For a video summary of this paper, please visit https://youtu.be/xJEpSBYoZJ0 . [ABSTRACT FROM AUTHOR]

Text A set A of nonnegative integers is said to be complete if every sufficiently large natural number is the sum of distinct terms taken from A . In 1959, Birch confirmed a conjecture of Erdős by proving that the set { p n q m : n , m = 0 , 1 , … } is complete, where p and q are two coprime integers greater than 1. In this paper, we study extensively the set P ( S p 1 , … , p r ) of all vectors ( m 1 , … , m r ) with integer coordinates which can be represented as ∑ ( p 1 α 1 , … , p r α r ) , where p 1 , … , p r are integers greater than 1 and α 1 , … , α r are nonnegative integers. We find many regular parts in P ( S p 1 , … , p r ) , for example, discrete rectangles. To study the problem, we find a new matching principle. Video For a video summary of this paper, please visit https://youtu.be/qXUe1E4jlLQ . [ABSTRACT FROM AUTHOR]

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture 1 is a claim about the uniqueness of certain monomial graphs. Conjecture 2 , which implies Conjecture 1 , deals with certain permutation polynomials of finite fields. Two natural strengthenings of Conjecture 2 , referred to as Conjectures A and B in the present paper, were also insinuated. In a recent development, Conjecture 2 and hence Conjecture 1 have been confirmed. The present paper gives a proof of Conjecture A . [ABSTRACT FROM AUTHOR]

Text Let k , l , a , b , c be positive integers such that gcd ⁡ ( k a , l b ) = 1 , min ⁡ { a , b , c } > 1 , a ≠ 3 , b ≠ 3 and 2 ∤ c . In this paper, we prove that there are at most four solutions in positive integers ( x , y , z ) to the equation k a x + l b y = c z and at most two solutions when 2 ∤ ( u ( l / k ) ) , where u ( m ) is the least positive integer t with m t ≡ 1 ( mod c ) . Video For a video summary of this paper, please visit https://youtu.be/Dt3Y7TDMxlg . [ABSTRACT FROM AUTHOR]

The N th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large N th linear complexity. However, in this paper we show that for q -automatic sequences over F q the converse is not true. We prove that any (not ultimately periodic) q -automatic sequence over F q has N th linear complexity of order of magnitude N . For some famous sequences including the Thue–Morse and Rudin–Shapiro sequence we determine the exact values of their N th linear complexities. These are non-trivial examples of predictable sequences with N th linear complexity of largest possible order of magnitude. [ABSTRACT FROM AUTHOR]

Text By Zeckendorf's theorem each positive integer is uniquely written as a sum of distinct non-adjacent terms of the Fibonacci sequence. This representability remains true for so called the Nth order Fibonacci sequence , and for a further generalization to linear recurrences with positive coefficients. In this paper we consider sequences { G n } that have the same linear recurrence relations as the N th order Fibonacci sequence but has different initial values, and investigate the number of positive integers up to X that are written as a sum of distinct terms of G n . We also introduce a converse of Zeckendorf's theorem that does not require the increasing condition. Our method extends to general linear recurrences, and a generalization is introduced in this paper. Video For a video summary of this paper, please visit https://youtu.be/vSwSJ_sppns . [ABSTRACT FROM AUTHOR]

Text A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors p 1 , … , p t of n and positive integers α 1 , … , α t such that n = p 1 α 1 + ⋯ + p t α t . It is clear that t ≥ 3 . In 1992, Erdős and Hegyvári proved that, for any prime p , there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by p . In this paper, we prove that, for any positive integer m , there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by m if and only if 8 ∤ m . We also present some related results and pose several problems for further research. Video For a video summary of this paper, please visit https://youtu.be/WC_VRFtY07c . [ABSTRACT FROM AUTHOR]

Text Let m be an integer with m ≥ 2 . For A ⊆ Z m and n ∈ Z m , let R 1 ( A , n ) , R 2 ( A , n ) , R 3 ( A , n ) denote the number of solutions of the equation a + a ′ = n with ordered pairs ( a , a ′ ) ∈ A × A , unordered pairs ( a , a ′ ) ∈ A × A ( a ≠ a ′ ) and unordered pairs ( a , a ′ ) ∈ A × A , respectively. In this paper, for i ∈ { 1 , 2 , 3 } , we determine all sets A , B ⊆ Z m such that R i ( A , n ) = R i ( B , n ) for all n ∈ Z m when the cardinality of the symmetric difference of A and B is small or large. These extend some previous results. We also pose some problems for further research. Video For a video summary of this paper, please visit https://youtu.be/stBa9Uy5U0I . [ABSTRACT FROM AUTHOR]

Text Two infinite sequences A and B of nonnegative integers are called additive complements , if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S = { 1 2 , 2 2 , … } . The following result is proved: if B = { b n } n = 1 ∞ with b n ≥ π 2 16 n 2 − 0.57 n 1 2 log ⁡ n − β n 1 2 for all positive integers n and any given constant β , then B is not an additive complement of S . In particular, B = { ⌊ π 2 16 n 2 ⌋ | n = 1 , 2 , … } is not an additive complement of S . Video For a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8 . [ABSTRACT FROM AUTHOR]

We prove a number of conjectures recently stated by P. Barry, related to the paperfolding sequence and the Rueppel sequence. Furthermore, we study the regularity of sequences involved in the paper, and prove that for all q ≥ 2 , the sequence consisting of the positive integers whose odd part is of the form 4 k + 1 is not q -regular. Finally we establish the 2-regularity of two sequences of Hankel determinants. [ABSTRACT FROM AUTHOR]

Text Let H n be the n -th harmonic number and let u n be its numerator. For any prime p , let J p be the set of positive integers n with p | u n . In 1991, Eswarathasan and Levine conjectured that J p is finite for any prime p . It is clear that the p -adic valuation of H n is not less than − ⌊ log p ⁡ n ⌋ . Let T p be the set of positive integers n such that the p -adic valuation of H n is equal to − ⌊ log p ⁡ n ⌋ . Recently, Carlo Sanna proved that | J p ∩ [ 1 , x ] | < 129 p 2 / 3 x 0.765 and that there exists S p ⊆ T p with δ ( S p ) > 0.273 , where δ ( X ) denotes the logarithmic density of the set X of positive integers. He also commented that with his methods δ ( S p ) > 1 / 3 − ε cannot be achieved. In this paper, we improve these results. For example, two of our results are: (a) | J p ∩ [ 1 , x ] | ≤ 3 x 2 / 3 + 1 / ( 25 log ⁡ p ) ; (b) δ ( T p ) exists and 1 − ( 2 log ⁡ p ) − 1 ≤ δ ( T p ) ≤ 1 − ( p log ⁡ p ) − 1 for all primes p ≥ 13 . In particular, δ ( T p ) > 0.63 for all primes p . Video For a video summary of this paper, please visit https://youtu.be/3ujCuVwH8k8 . [ABSTRACT FROM AUTHOR]

In a prequel to this paper [1] a notion of curvature on the integers was introduced, based on a formal patching technique. In this paper, which is essentially independent of its prequel, we introduce another notion of curvature on the integers, based on “algebraization of Frobenius lifts by correspondences.” Our main results are vanishing/non-vanishing theorems for this new type of curvature in the case of “Chern connections” attached to classical groups. [ABSTRACT FROM AUTHOR]