Search

The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length n p s over p m , where p is a prime number and p does not divide n , the problem has been solved completely for all n ≤ 6 and partially for n = 7 , 8. In this paper, we solve the problem for n = 9 and all primes p different from 3 and 1 9. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length 9 p s over p m . As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the skew polynomial ring R = ( p + u p) [ x ; ] and its quotient ring R n = R / 〈 x n − 1 〉 , where p is an odd prime number, u 2 = 0 , and (u) = − u. We give an explicit structure of the ideals in R and R n and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian. [ABSTRACT FROM AUTHOR]

Since simple commutative finite flat group schemes over Z are killed by a prime number p , their order is a power of p. Abraškin and Fontaine have both shown that for primes p ⩽ 17 the only simple p -power order group schemes are μ p and Z / p Z. We extend their result to p = 19. [ABSTRACT FROM AUTHOR]

Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus Q is a polynomial in q [ T ] such that Q (0) ≠ 0. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition. [ABSTRACT FROM AUTHOR]

In the field of Cryptography, Quantum Key Distribution (QKD) is an application of Quantum Information theory that obtained a great deal of attention in recent years. It allows to establish secret keys between two or more parties, in a much safer way than that implemented by classic cryptography (based on discrete logarithms and factorization of prime numbers). The most promising way of realizing a QKD network (especially over great distances) in the near future is by a constellation of satellites. This paper considers the problem of optimizing the orbits of the satellites in order to maximize the minimum key length shared in a network of ground stations over a fixed amount of time. Different networks of stations are considered and the influence of their geographical disposition on the design and the performance index is highlighted. The networks considered are: a global constellation, a regional European constellation, and two in which there are groups of stations in two different narrow bands of latitude. The effect of Inter-satellite links is then taken into account and how, in some cases, they can improve the performances. Finally the daily performance of the considered constellations are analyzed. [ABSTRACT FROM AUTHOR]

Let f (z) = n F n − 1 (α , β) be the hypergeometric series with parameters α = (α 1 , ... , α n) and β = (β 1 , ... , β n − 1 , 1) in (Q ∩ (0 , 1 ]) n , let d α , β be the least common multiple of the denominators of α 1 , ... , α n , β 1 , ... , β n − 1 written in lowest form and let p be a prime number such that p does not divide d α , β and f (z) ∈ Z (p) [ [ z ] ]. Recently in [11] , it was shown that if for all i , j ∈ { 1 , ... , n } , α i − β j ∉ Z then the reduction of f (z) modulo p is algebraic over F p (z). A standard way to measure the complexity of an algebraic power series is to estimate its degree and its height. In this work, we prove that if p > 2 d α , β then there is a nonzero polynomial P p (Y) ∈ F p (z) [ Y ] having degree at most p 2 n φ (d α , β) and height at most 5 n (n + 1) ! p 2 n φ (d α , β) such that P p (f (z) mod p) = 0 , where φ is the Euler's totient function. Furthermore, our method of proof provides us a way to make an explicit construction of the polynomial P p (Y). We illustrate this construction by applying it to some explicit hypergeometric series. [ABSTRACT FROM AUTHOR]

For arbitrary real t > 1 we examine the set { ⌊ x / n t ⌋ : n ≤ x }. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set. [ABSTRACT FROM AUTHOR]

Let p be a prime number. A saturated fusion system F on a finite p -group S is said to be supersolvable if there is a series 1 = S 0 ≤ S 1 ≤ ... ≤ S m = S of subgroups of S such that S i is strongly F -closed for all 0 ≤ i ≤ m and such that S i + 1 / S i is cyclic for all 0 ≤ i < m. We prove some criteria that ensure that a saturated fusion system F on a finite p -group S is supersolvable provided that certain subgroups of S are abelian and weakly F -closed. Our results can be regarded as generalizations of purely group-theoretic results of Asaad [3]. [ABSTRACT FROM AUTHOR]

Given an odd prime number p and a p-stabilized Artin representation \rho over \mathbb {Q}, we introduce a family of p-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a p-adic Stark conjecture which can be seen as an explicit strengthening of conjectures by Perrin-Riou and Benois in the context of Artin motives. We show that these conjectures imply the p-part of the Tamagawa number conjecture for Artin motives at s=0 and we obtain unconditional results on the torsionness of Selmer groups. We also relate our new conjectures with various main conjectures and variants of p-adic Stark conjectures that appear in the literature. In the case of monomial representations, we prove that our conjectures are essentially equivalent to some newly introduced Iwasawa-theoretic conjectures for Rubin-Stark elements. We derive from this a p-adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which p is inert. Along the way, we prove that the Gross-Kuz'min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

Let p$p$ be a prime number, let g(x)=xp2+pr+2xp2+1$g(x)=x^{p^{2}}+p^{r+2}x^{p^{2}+1}$ with r∈Z⩾0$r\in \mathbb {Z}_{\geqslant 0}$, and let ϕ(x)=x+O(x2)$\phi (x)=x+O(x^{2})$ be the Böttcher coordinate satisfying ϕ(g(x))=ϕ(x)p2$\phi (g(x))=\phi (x)^{p^{2}}$. Salerno and Silverman conjectured that the radius of convergence of ϕ−1(x)$\phi ^{-1}(x)$ in Cp$\mathbb {C}_{p}$ is p−p−r/(p−1)$p^{-p^{-r}/(p-1)}$. In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result. [ABSTRACT FROM AUTHOR]

Let p⩾5$p\geqslant 5$ be a prime number. We consider the Iwasawa λ$\lambda$‐invariants associated to modular Bloch–Kato Selmer groups, considered over the cyclotomic Zp$\mathbb {Z}_p$‐extension of Q$\mathbb {Q}$. Let g$g$ be a p$p$‐ordinary cuspidal newform of weight 2 and trivial nebentype. We assume that the μ$\mu$‐invariant of g$g$ vanishes, and that the image of the residual representation associated to g$g$ is suitably large. We show that for any number n$n$ greater than or equal to the λ$\lambda$‐invariant of g$g$, there are infinitely many newforms f$f$ that are p$p$‐congruent to g$g$, with λ$\lambda$‐invariant equal to n$n$. We also prove quantitative results regarding the levels of such modular forms with prescribed λ$\lambda$‐invariant. [ABSTRACT FROM AUTHOR]

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants p , n , s , and t , where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of p 6 and p 7 that hold substantial importance in the field of coding theory. [ABSTRACT FROM AUTHOR]

The technique of distinguishing one knot from another by coloring arcs and applying some basic modular arithmetic is part of most standard undergraduate knot theory classes. When we study n-colorability, we are usually only interested when n is a prime number. But what if n is composite? What can we say then? [ABSTRACT FROM AUTHOR]

Let 휎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 휎-subnormal subgroup is modular. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms f and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals. [ABSTRACT FROM AUTHOR]

Consider the family of elliptic curves |$E_{n}:y^{2}=x^{3}+n^{2}$|⁠ , where |$n$| varies over positive cubefree integers. There is a rational |$3$| -isogeny |$\phi $| from |$E_{n}$| to |$\hat {E}_{n}:y^{2}=x^{3}-27n^{2}$| and a dual isogeny |$\hat {\phi }:\hat {E}_{n}\rightarrow E_{n}$|⁠. We show that for almost all |$n$|⁠ , the rank of |$\operatorname {Sel}_{\phi }(E_{n})$| is |$0$|⁠ , and the rank of |$\operatorname {Sel}_{\hat {\phi }}(\hat {E}_{n})$| is determined by the number of prime factors of |$n$| that are congruent to |$2\bmod 3$| and the congruence class of |$n\bmod 9$|⁠. [ABSTRACT FROM AUTHOR]

We consider cyclic algebras, Milnor's symbol algebras, and certain graded algebra structures on them. We classify these gradings with respect to both isomorphism and equivalence relations. Some of them induce gradings on matrix algebras, which we also classify. As an application, we obtain the classification of all group gradings on the algebra M p (F) , where p is a prime number and F is an arbitrary field. [ABSTRACT FROM AUTHOR]

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index n ≥ 49. To investigate how the nodes are connected, we analytically describe that the link density saturates to 6 / π 2 , whereas the average degree increases linearly with slope 6 / π 2 with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for 49 ≤ n ≤ 288 and it is at most 2 for n ≥ 289. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of r -length labeled cycles, which indicates the existence of a cycle of length at most O (log n). Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős–Rényi random network and Barabási–Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator–prey networks which can easily avoid the global synchronization. [ABSTRACT FROM AUTHOR]

In this paper, we establish a crucial requirement for a number of the form n , having two prime factors p and q such that (p , q) ≡ (1 , 3) (mod 8) , to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field Q (− 2 p q) when n is congruent. [ABSTRACT FROM AUTHOR]

For a function f : N → N , let N f + (x) = { n ⩽ x : n = k + f (k) for some k }. Let τ (n) = ∑ d | n 1 be the divisor function, ω (n) = ∑ p | n 1 be the prime divisor function, and φ (n) = # { 1 ⩽ k ⩽ n : gcd ⁡ (k , n) = 1 } be Euler's totient function. We show that (1) x ≪ N ω + (x) , (2) x ≪ N τ + (x) ⩽ 0.94 x , (3) x ≪ N φ + (x) ⩽ 0.93 x. [ABSTRACT FROM AUTHOR]

For a prime number p and an integer m ≥ 2 , we prove that the symbol length of all elements of m -fold Massey products in H 2 (G , F p) , for pro- p groups G of elementary type, is bounded by (m 2 / 4) + m. Assuming the Elementary Type Conjecture, this applies to all finitely generated maximal pro- p Galois groups G = G F (p) of fields F which contain a root of unity of order p. More generally, we provide such a uniform bound for the symbol length of all pullbacks ρ ⁎ (ω ¯) of a given cohomology element ω ¯ ∈ H n (G ¯ , F p) , where G ¯ is a finite p -group, n ≥ 2 , and ρ : G → G ¯ is a pro- p group homomorphism. [ABSTRACT FROM AUTHOR]

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences (u n) n ≥ 0 in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes u n , or else there exists a constant c u > 0 (which we can give good approximations to) such that there are ∼ c u log ⁡ N primes u n with n ≤ N , as N → ∞. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the u n (mod p). [ABSTRACT FROM AUTHOR]

For a prime number |$\ell $|⁠ , we introduce and study oriented right-angled Artin pro- |$\ell $| groups |$G_{\Gamma ,\lambda }$| (oriented pro- |$\ell $| RAAGs for short) associated to a finite oriented graph |$\Gamma $| and a continuous group homomorphism |$\lambda \colon{\mathbb{Z}}_{\ell }\to{\mathbb{Z}}_{\ell }^{\times }$|⁠. We show that an oriented pro- |$\ell $| RAAG |$G_{\Gamma ,\lambda }$| is a Bloch–Kato pro- |$\ell $| group if, and only if, |$(G_{\Gamma ,\lambda },\theta _{\Gamma ,\lambda })$| is an oriented pro- |$\ell $| group of elementary type, generalizing a recent result of I. Snopce and P. Zalesskiĭ—here |$\theta _{\Gamma ,\lambda }\colon G_{\Gamma ,\lambda }\to{\mathbb{Z}}_{\ell}^{\times }$| denotes the canonical |$\ell $| -orientation on |$G_{\Gamma ,\lambda }$|⁠. This yields a plethora of new examples of pro- |$\ell $| groups that are not maximal pro- |$\ell $| Galois groups. We invest some effort in order to show that oriented right-angled Artin pro- |$\ell $| groups share many properties with right-angled Artin pro- |$\ell $| -groups or even discrete RAAG's, for example, if |$\Gamma $| is a specially oriented chordal graph, then |$G_{\Gamma ,\lambda }$| is coherent generalizing a result of C. Droms. Moreover, in this case, |$(G_{\Gamma ,\lambda },\theta _{\Gamma ,\lambda })$| has the Positselski–Bogomolov property generalizing a result of H. Servatius, C. Droms, and B. Servatius for discrete RAAG's. If |$\Gamma $| is a specially oriented chordal graph and |$\operatorname{Im}(\lambda)\subseteq 1+4{\mathbb{Z}}_{2}$| in case that |$\ell =2$|⁠ , then |$H^{\bullet }(G_{\Gamma ,\lambda },{\mathbb{F}}_{\ell }) \simeq \Lambda ^{\bullet }(\ddot{\Gamma }^{\textrm{op}})$| generalizing a well-known result of M. Salvetti (cf. [ 39 ]). Dedicated to the memory of Avinoam Mann. [ABSTRACT FROM AUTHOR]

Let E/Q$E/\mathbb {Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod ℓ$\ell$ Galois representation ρ¯E,ℓ$\overline{\rho }_{E, \ell }$ of E$E$ is surjective for each prime number ℓ$\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime ℓ$\ell$, linear in the logarithm of the conductor of E$E$, such that ρ¯E,ℓ$\overline{\rho }_{E, \ell }$ is nonsurjective. [ABSTRACT FROM AUTHOR]

Many studies have shown that the brain can process subliminal numerals, i.e., participants can categorize a subliminal number into two categories: greater than 5 or less than 5. In the context of many studies on the unconscious integration of multiple subliminal stimuli, the issue of whether multiple subliminal numbers can be integrated is contentious. The same-different task is regarded as a perfect tool to explore unconscious integration. In the two experiments reported, we used a same-different task in which a pair of masked prime numbers was followed by a pair of target numbers, and participants were asked to decide whether the two target numbers were on the same (both smaller or larger than 5) or different sides (one smaller, the other larger than 5) of 5 in magnitude. The results indicated that the prime numbers could be categorized unconsciously, which was reflected by the category priming effect, and that the unconscious category relationship of the two prime numbers could affect the judgment on the category relationship of the two target numbers, as reflected by the response priming effect. The duration of the prime-to-target interstimulus interval (ISI) was also manipulated, showing a positive compatibility effect (PCE) of category priming and a negative compatibility effect (NCE) of response priming no matter whether the ISI was short (50 ms) or long (150 ms). The NCE, which occurred when the prime-to-target ISI was relatively short in this study, contradicted the conventional view but was consistent with previous results of unconscious integration based on an attention modulation mechanism. Importantly, this study provided evidence for the still-under-debate issue of numerical information integration. [ABSTRACT FROM AUTHOR]

Fix a prime number p and let E / F be a CM extension of number fields in which p splits relatively. Let π be an automorphic representation of a quasi-split unitary group of even rank with respect to E / F such that π is ordinary above p with respect to the Siegel parabolic subgroup. We construct the cyclotomic p -adic L -function of π , and a certain generating series of Selmer classes of special cycles on Shimura varieties. We show, under some conditions, that if the vanishing order of the p -adic L -function is 1, then our generating series is modular and yields explicit nonzero classes (called Selmer theta lifts) in the Selmer group of the Galois representation of E associated with π ; in particular, the rank of this Selmer group is at least 1. In fact, we prove a precise formula relating the p -adic heights of Selmer theta lifts to the derivative of the p -adic L -function. In parallel to Perrin-Riou's p -adic analogue of the Gross–Zagier formula, our formula is the p -adic analogue of the arithmetic inner product formula recently established by Chao Li and the second author. [ABSTRACT FROM AUTHOR]

Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ (A) , the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2 ℓ (A) if | G | is odd. More generally we prove that the nilpotent length of G is at most 2 ℓ (A) + c (G ; A) when G is of odd order and A normalizes a Sylow system of G where c (G ; A) denotes the number of trivial A -modules appearing in an A -composition series of G. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number, q = p m for a positive integer m , let q be the finite field with q elements and ω be a primitive element of q . We first give an orthogonal decomposition of the ring R = q + ν q , where ν 2 = a 3 , and a = ω 2 l for a fixed integer l. In addition, Galois dual of a linear code over R is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring R are investigated as well. Remarkably, we obtain that if linear codes and are a complementary pair, then the code and the dual code ⊥ E of are equivalent to each other. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime number. We classify the subgroups G of Sp 4 (F ℓ) and GSp 4 (F ℓ) that act irreducibly on F ℓ 4 , but such that every element of G fixes an F ℓ -vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree ℓ between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and ℓ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ℓ for which some abelian surface A / Q fails the local-global principle for isogenies of degree ℓ . [ABSTRACT FROM AUTHOR]

Traditional two-color lights out games and their solutions have been studied extensively. This paper focuses on the existence, or lack there of, of solutions in more general lights out games, which take on a prime number of color states. The result in this paper generalizes the existence of solutions in the family of lights out games on a (p + 1) × n grid with p -colors where n ∈ N and p. [ABSTRACT FROM AUTHOR]

The systematic generation of prime numbers has been almost ignored since the 1990s, when most of the IT research resources related to prime numbers migrated to studies on the use of very large primes for cryptography, and little effort was made to further the knowledge regarding techniques like sieving. At present, sieving techniques are mostly used for didactic purposes, and no real advances seem to be made in this domain. This systematic review analyzes the theoretical advances in sieving that have occurred up to the present. The research followed the PRISMA 2020 guidelines and was conducted using three established databases: Web of Science, IEEE Xplore and Scopus. Our methodical review aims to provide an extensive overview of the progress in prime sieving—unfortunately, no significant advancements in this field were identified in the last 20 years. [ABSTRACT FROM AUTHOR]

We found a lower bound on the rank of a square matrix where every entry in the leading diagonal is neither zero nor one and every entry outside the leading diagonal is either zero or one. The rank of this matrix is at least half its order. Under an additional condition, the lower bound is higher by one. This condition means that some auxiliary system of linear equations has no binary solution. Some examples are provided that show that the lower bound can be achieved. This lower bound on the matrix rank allows the problem of finding a binary solution to a system of linear equations with a sufficiently large number of linearly independent equations to be reduced to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each differing from the binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a large system of linear algebraic equations. Estimates of the time required for calculating the matrix rank in the SymPy computer algebra system are also provided. It is shown that the rank of a matrix over the field of residues modulo prime number is calculated faster than it generally takes to calculate the rank of a matrix of the same order over the field of rational numbers. [ABSTRACT FROM AUTHOR]

Denote by τ k (n) , ω (n) and μ 2 (n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [ t ] be the integral part of real number t. For f = ω , 2 ω , μ 2 , τ k , we prove that ∑ n ≤ x f x n = x ∑ d ≥ 1 f (d) d (d + 1) + O (x f + ) for x → ∞ , where ω = 5 3 1 1 0 , 2 ω = 9 1 9 , μ 2 = 2 5 , τ k = 5 k − 1 1 0 k − 1 and > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès. [ABSTRACT FROM AUTHOR]

Let p be a prime such that p ≡ 3 (mod 4). Then, we show that there is no imaginary cyclic quartic extension K of ℚ whose class number is p. Suppose L / ℚ is a cyclic extension of number fields with an odd degree. Then, we show that 2 does not divide the class number of L if the class group of L is cyclic. We also construct some families of number fields whose class number is not divisible by a fixed prime. [ABSTRACT FROM AUTHOR]

Let p$p$ be a prime number and m$m$ an integer coprime to p$p$. In the spirit of arithmetic topology, we introduce the notions of the twisted Iwasawa invariants λ,μ,ν$\lambda , \mu , \nu$ of GLN${\rm GL}_N$‐representations and Z/mZ×Zp${\mathbb {Z}}/m{\mathbb {Z}}\times {\mathbb {Z}}_{p}$‐covers of knots. We prove among other things that the set of Iwasawa invariants determines the genus and the fiberedness of a knot, yielding their profinite rigidity. Several intuitive examples are attached. We further prove the μ=0$\mu =0$ theorem for SL2${\rm SL}_2$‐representations of twist knot groups and give some remarks. [ABSTRACT FROM AUTHOR]

Let |$\mathbb{F}_{q}$| be a finite field whose characteristic is relatively prime to |$2$| and |$3$|⁠. Let |$p$| be a prime number that is coprime to |$q$|⁠. Let |$E$| be an elliptic curve over the global function field |$K = \mathbb{F}_{q}(t)$| such that |$\textrm{Gal}(K(E[p])/K)$| contains the special linear group |$\textrm{SL}_{2}(\mathbb{F}_{p})$|⁠. We show that if the quadratic twist family of |$E$| has an element whose Néron model has a multiplicative reduction away from |$\infty $|⁠ , then the average |$p$| -Selmer rank is |$p+1$| in large |$q$| -limit for almost all primes |$p$|⁠. [ABSTRACT FROM AUTHOR]

Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $ -towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $\ell $ and the abelian $\ell $ -tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $\mu $ and $\lambda $ invariants. [ABSTRACT FROM AUTHOR]

This article corrects a mistake in 'Explicit interval estimates for prime numbers', Math. Comp. 91 (2022), 1955–1970. The error was in a closed-form expression for an integral involving the smoothing function. The table of pairs of (\Delta,x_0) is recomputed for the main theorem, which states that for all x \geq x_0 there exists at least one prime in the interval (x(1-\Delta ^{-1}),x]. [ABSTRACT FROM AUTHOR]

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd k ≤ 10 5 and even k ≤ 2 ⋅ 10 5 that any even integer n ≥ 40 can be expressed as the sum of a prime and a square-free number coprime to k. We also discuss applications to other Goldbach-like problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms 2 a 3 b and 2 c 5 d. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem. [ABSTRACT FROM AUTHOR]

We prove that modular categories of Frobenius–Perron dimension p 2 q 2 r 2 are solvable, where p < q < r are prime numbers. As applications, we get that integral modular categories of Frobenius–Perron dimension less than 1 8 0 0 are solvable, and hence integral perfect modular categories have Frobenius–Perron dimension greater than or equal to 1 8 0 0. When the modular categories considered are weakly group-theoretical, we further prove that integral perfect modular categories have Frobenius–Perron dimension greater than or equal to 3 6 0 0. [ABSTRACT FROM AUTHOR]

For an n$n$‐bit positive integer a$a$ written in binary as a=∑j=0n−1εj(a)2j,$ {a} = \sum _{j=0}^{n-1} \varepsilon _{j}(a) \,2^j,$ where εj(a)∈{0,1}$\varepsilon _j(a) \in \lbrace 0,1\rbrace$, j∈{0,...,n−1}$j\in \lbrace 0, \ldots , n-1\rbrace$, εn−1(a)=1$\varepsilon _{n-1}(a)=1$, let us define a←=∑j=0n−1εj(a)2n−1−j,$ \overleftarrow{a} = \sum _{j=0}^{n-1} \varepsilon _j(a)\,2^{n-1-j},$ the digital reversal of a$a$. Also let Bn={2n−1⩽a<2n:aodd}${\mathcal {B}}_n= \lbrace 2^{n-1}\leqslant a<2^n:\nobreakspace a \text{ odd}\rbrace$. With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of p∈Bn$p \in {\mathcal {B}}_n$ such that p$p$ and p←$\overleftarrow{p}$ are prime. We also prove that for sufficiently large n$n$, {a∈Bn:max{Ω(a),Ω(a←)}⩽8}⩾c2nn2,$$\begin{equation*} \hspace*{30pt}{\left| \lbrace a \in {\mathcal {B}}_n:\nobreakspace \max \lbrace \Omega (a), \Omega (\overleftarrow{a})\rbrace \leqslant 8 \rbrace \right|} \geqslant c\, \frac{2^n}{n^2}, \end{equation*}$$where Ω(n)$\Omega (n)$ denotes the number of prime factors counted with multiplicity of n$n$ and c>0$c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of n$n$‐bit integers a$a$ such that a$a$ and a←$\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum ∑a∈Bnexp2πi(αa+ϑa←)(α,ϑ∈R).$$\begin{equation*} \hspace*{30pt}\sum _{a \in {\mathcal {B}}_n} \exp {\left(2\pi i (\alpha a + \vartheta \overleftarrow{a})\right)} \quad (\alpha,\vartheta \in \mathbb {R}). \end{equation*}$$ [ABSTRACT FROM AUTHOR]

A classical problem due to Abel is to determine if a differential equation |$y^{\prime}=\eta y$| admits a non-trivial solution |$y$| algebraic over |$\mathbb C(x)$| when |$\eta $| is a given algebraic function over |$\mathbb C(x)$|⁠. Risch designed an algorithm that, given |$\eta $|⁠ , determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when |$\eta $| admits a Puiseux expansion with rational coefficients at some point in |$\mathbb C\cup \{\infty \}$|⁠ , which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of |$y^{\prime}=\eta y$| if and only if the coefficients of the Puiseux expansion of |$x\eta (x)$| at |$0$| satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations |$y^{\prime}=\eta y$| with an algebraic solution when |$x\eta (x)$| is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks. [ABSTRACT FROM AUTHOR]

The quantum degeneracies of Bogomolny‐Prasad‐Sommerfield (BPS) black holes of octonionic magical supergravity in five dimensions are studied. Quantum degeneracy is defined purely number theoretically as the number of distinct states in charge space with a given set of invariant labels. Quantum degeneracies of spherically symmetric stationary BPS black holes are given by the Fourier coefficients of modular forms of exceptional group E7(−25)$E_{7(-25)}$. Their charges take values in the lattice defined by the exceptional Jordan algebra over integral octonions. The quantum degeneracies of rank 1 and rank 2 BPS black holes are given by the Fourier coefficients of singular modular forms E4(Z)$E_4(Z)$ and E8(Z)$E_8(Z)$. The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on embeddings of cubic rings A into the exceptional Jordan algebra we show that the quantum degeneracies of rank 1 black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms (HMFs) of SL(2,A)$SL(2,A)$. If the discriminant of the cubic ring A is D=p2$D=p^2$ with p a prime number then the isotropic lines in the 24 dimensional orthogonal complement of A define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. The current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity is also reviewed. [ABSTRACT FROM AUTHOR]

The document titled "Problems and Solutions" is an article from the American Mathematical Monthly that provides a list of proposed mathematical problems along with their solutions. The problems cover various topics and are contributed by different mathematicians from around the world. The article also includes instructions for submitting proposed problems and solutions. Additionally, it features a solution to a specific problem related to logarithmic trigonometric integrals. The given text presents a mathematical equation and its evaluation. The equation involves integrals and substitutions, and it is used to calculate a value called "I." The text provides step-by-step calculations and uses mathematical identities to simplify the equation. The final result is that I equals a specific value. The text also mentions the use of Fourier sine series in the calculations. The given text contains mathematical formulas and solutions to mathematical problems. The first part of the text presents a formula for the integral of a sine function, depending on whether the value of n is even or odd. The second part discusses the solution to a problem involving operator norms. The solution involves applying the Gram-Schmidt process to a basis in a certain space and obtaining an orthonormal basis. The text also mentions the names of individuals and groups who have solved the problems. The given text discusses the properties of a function f that belongs to a set S. It states that for any function f in S, the sum of the squares of the values of f and its derivative is equal to a constant. It also mentions that the supremum of [Extracted from the article]

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]