TY - JOUR AU - Bertók, Csanád AU - Hajdu, Lajos TI - The resolution of three exponential Diophantine equations in several variables JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 260 PY - 2024 SP - 29 EP - 40 PG - 12 SN - 0022-314X DO - 10.1016/j.jnt.2024.01.009 UR - https://m2.mtmt.hu/api/publication/34781671 ID - 34781671 LA - English DB - MTMT ER - TY - JOUR AU - Long, Hanqing AU - Wei, Dasheng TI - The 3-rd unramified cohomology for norm one torus JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 257 PY - 2024 SP - 186 EP - 201 PG - 16 SN - 0022-314X DO - 10.1016/j.jnt.2023.10.014 UR - https://m2.mtmt.hu/api/publication/34616343 ID - 34616343 AB - For an algebraic torus S, Blinstein and Merkurjev have given an estimate of 3-rd unramified cohomology (H) over bar (3)(nr)(F(S), Q/Z(2)) obtained from a flasque resolution of S. Based on their work, for the norm one torus W = R(K/F)((1))G(m) with K/F abelian, we compute the 3-rd unramified cohomology (H) over bar (3)(nr)(F(W), Q/Z(2)). (c) 2023 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Fink, Thomas M. A. TI - Recursively divisible numbers JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 256 PY - 2024 SP - 37 EP - 54 PG - 18 SN - 0022-314X DO - 10.1016/j.jnt.2023.08.008 UR - https://m2.mtmt.hu/api/publication/34602443 ID - 34602443 AB - We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: kappa(x)(n) = n(x)+ Sigma(dleft perpendicularn) kappa(x)(d), where the sum is over the proper divisors of n. We give a geometrical interpretation of kappa(x)(n), which we use to derive a relation between kappa(x)(n) and kappa(0)(n). For x >= 2, we observe that kappa(x)(n)/n(x) < 1/(2 - zeta(x)). We show that, for n >= 2, kappa(0)(n) is twice the number of ordered factorizations, a problem much studied in its own right. By computing those numbers that are more recursively divisible than all of their predecessors, we recover many of the numbers prevalent in design and technology, and suggest new ones which have yet to be adopted. (c) 2023 The Author(s). Published by Elsevier Inc. LA - English DB - MTMT ER - TY - JOUR AU - Li, Rao AU - Lu, Fan TI - Run-length function of the beta-expansion of a fixed real number JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 256 PY - 2024 SP - 55 EP - 78 PG - 24 SN - 0022-314X DO - 10.1016/j.jnt.2023.09.002 UR - https://m2.mtmt.hu/api/publication/34602431 ID - 34602431 N1 - Funding Agency and Grant Number: NSFC [12271382]; Science and Technology Department of Sichuan Province [2021JDJQ0030, 2022JDTD0019] Funding text: This work was supported by NSFC No. 12271382 and Science and Technology Department of Sichuan Province Nos. 2021JDJQ0030, 2022JDTD0019. AB - For any real numbers x is an element of [0, 1] and beta > 1, let r(n)(x, beta) be the maximal length of consecutive 0's in the first n digits of the beta-expansion of x in base beta. The run-length function r(n)(x, beta) has been well studied for a fixed base beta > 1 or a fixed real number x = 1. In this paper, we prove that for any x is an element of (0, 1), the setD-x = {beta > 1: lim(n ->infinity) r(n)(x, beta)/log(beta) n = 1}is of full Lebesgue measure in (1, +infinity). When the exceptional set is considered, we prove that for any real numbers 0 <= a <= b <= + infinity, the setD-x = {beta > 1: lim inf(n ->infinity) r(n)(x, beta)/log(beta) n = a,lim sup(n ->infinity) r(n)(x, beta)/log(beta) n = b,is of full Hausdorff dimension. We also determine the Hausdorff dimension of the setF-x(c, d) = {beta > 1 : lim inf(n ->infinity) r(n)(x, beta)/n = c,lim sup(n ->infinity) r(n)(x, beta/n = d}for any real numbers 0 <= c <= d <= 1. (c) 2023 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Ren, Xiumin AU - Zhang, Qingqing AU - Zhang, Rui TI - Roth-type theorem for quadratic system in Piatetski-Shapiro primes JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 257 PY - 2024 SP - 1 EP - 23 PG - 23 SN - 0022-314X DO - 10.1016/j.jnt.2023.10.012 UR - https://m2.mtmt.hu/api/publication/34584871 ID - 34584871 AB - Let c(1), . . . , c(s) be nonzero integers satisfying c(1) + + c(s) = 0. We consider the rational quadratic system c(1)x(1)(2)+ +c(s)x(s)(2) = 0 where x(i) are restricted in subset A of Piatetski-Shapiro primes not exceeding x and corresponding to c. We show that for c is an element of ( 1, min{ s/ s-1 , 29/ 28 } ) , if the system has only K -trivial solutions in A, then |A| < x(1/c)(log x)(-1) (log log log log x)((2-s)/(2c)+epsilon) holds for s >= 7.(c) 2023 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Lu, Dawei AU - Wang, Ruoyi TI - Optimizing the coefficients of the Ramanujan expansion JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 257 PY - 2024 SP - 146 EP - 162 PG - 17 SN - 0022-314X DO - 10.1016/j.jnt.2023.10.010 UR - https://m2.mtmt.hu/api/publication/34412755 ID - 34412755 LA - English DB - MTMT ER - TY - JOUR AU - Charlier, Emilie AU - Cisternino, Celia AU - Kreczman, Savinien TI - On periodic alternate base expansions JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 254 PY - 2024 SP - 184 EP - 198 PG - 15 SN - 0022-314X DO - 10.1016/j.jnt.2023.07.008 UR - https://m2.mtmt.hu/api/publication/34268081 ID - 34268081 N1 - Funding Agency and Grant Number: FNRS [J.0034.22]; FNRS Research Fellow grant 1 [1.A.789.23F, 1.A.564.19F] Funding text: We thank the referee for helpful suggestions. Emilie Charlier is supported by the FNRS grant J.0034.22. Celia Cisternino is supported by the FNRS Research Fellow grant 1.A.564.19F. Savinien Kreczman is supported by the FNRS Research Fellow grant 1.A.789.23F. AB - For an alternate base 0 = (& beta;0, . . . , & beta;p-1), we show that if all rational numbers in the unit interval [0, 1) have periodic expansions with respect to the p shifts of 0, then the bases & beta;0, . . . , & beta;p-1 all belong to the extension field Q(& beta;) where & beta; is the product & beta;0 & BULL; & BULL; & BULL; & beta;p-1 and moreover, this product & beta; must be either a Pisot number or a Salem number. We also prove the stronger statement that if the bases & beta;0, . . . , & beta;p-1 belong to Q(& beta;) but the product & beta; is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic 0-expansion is nowhere dense in [0, 1). Moreover, in the case where the product & beta; is a Pisot number and the bases & beta;0, ... , & beta;p-1 all belong to Q(& beta;), we prove that the set of points in [0, 1) having an ultimately periodic 0 -expansion is precisely the set Q(& beta;) & AND; [0, 1). For the restricted case of Renyi real bases, i.e., for p = 1 in our setting, our method gives rise to an elementary proof of Schmidt's original result. Therefore, even though our results generalize those of Schmidt, our proofs should not be seen as generalizations of Schmidt's original arguments but as an original method in the generalized framework of alternate bases, which moreover gives a new elementary proof of Schmidt's results from 1980. As an application of our results, we show that if 0 = (& beta;0, . . . ,& beta;p-1) is an alternate base such that the product & beta; of the bases is a Pisot number and & beta;0, . . . , & beta;p-1 & ISIN; Q(& beta;), then 0 is a Parry alternate base, meaning that the quasi-greedy expansions of 1 with respect to the p shifts of the base 0 are ultimately periodic.& COPY; 2023 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Komornik, Vilmos AU - Loreti, Paola AU - Pedicini, Marco TI - A quasi-ergodic approach to non-integer base expansions JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 254 PY - 2024 SP - 146 EP - 168 PG - 23 SN - 0022-314X DO - 10.1016/j.jnt.2023.07.009 UR - https://m2.mtmt.hu/api/publication/34268080 ID - 34268080 AB - P. Erdos et al. proved in 1990 that every nontrivial number has a continuum of expansions on two-letter alphabets in every base smaller than the Golden ratio, and that this property fails for the Golden ratio base. It was shown in a recent paper of Baiocchi et al. that if we replace the powers of the Golden ratio by the closely related Fibonacci sequence, then the resulting integer base expansions still have the continuum expansion property. The proof heavily relied on the special properties of the Golden ratio. The difficulty came from the fact that the new expansions do not have any more the ergodic structure of non-integer base expansions. In this paper we introduce a new "quasi-ergo dic" approach that allows us to handle many more general cases. We apply this approach to Baker's generalized Golden ratios.& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/). LA - English DB - MTMT ER - TY - JOUR AU - Pasten, Hector TI - Shimura curves and the abc conjecture JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 254 PY - 2024 SP - 214 EP - 335 PG - 122 SN - 0022-314X DO - 10.1016/j.jnt.2023.07.002 UR - https://m2.mtmt.hu/api/publication/34236588 ID - 34236588 AB - In this work we develop a framework that enables the use of Shimura curve parametrizations of elliptic curves to approach the abc conjecture, leading to a number of new unconditional applications over Q and, more generally, totally real number fields. Several results of independent interest are obtained along the way, such as bounds for the Manin constant, a study of the congruence number, extensions of the Ribet-Takahashi formula, and lower bounds for the L2-norm of integral quaternionic modular forms.The methods require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations.& COPY; 2023 Published by Elsevier Inc. LA - English DB - MTMT ER - TY - JOUR AU - Kumchev, Angel AU - Mccormick, Wade AU - Mcnew, Nathan AU - Park, Ariana AU - Scherr, Russell AU - Ziehr, Willow TI - Explicit bounds for large gaps between squarefree integers JF - JOURNAL OF NUMBER THEORY J2 - J NUMBER THEORY VL - 254 PY - 2024 SP - 336 EP - 357 PG - 22 SN - 0022-314X DO - 10.1016/j.jnt.2023.07.003 UR - https://m2.mtmt.hu/api/publication/34232092 ID - 34232092 AB - We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any x & GE; 2, that every interval of the form (x, x + 11x1/5 log x] contains a squarefree integer. The constant 11 can be improved further, if x is assumed to be larger than a (very) large constant.& COPY; 2023 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER -