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The Mathematics Student ISSN: 0025-5742 Vol. 91, Nos. 1-2, January-June (2022), 225{232 PROBLEM SECTION, Problems M1,M2 and M3
Abstract
Mathematics Problems
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Let $n\geq1$ be an integer. A magic square of order $n$ is a square table $n\times n$, say $A$, filled with distinct positive numbers $1,2,\ldots,n^2$ such that all cells of $A$ are distinct and the sum of the numbers in each row, column and diagonal is equal.Let $M(n,s)$ be the set of all $n\times n$ $(0,1)$-matrices, say $T$, such that the number of $1$ in every row and every column of $T$ is $s$.In this paper for every positive integer $k$ we find a new way for constructing magic squares of order $4k$. We show that the number of magic squares of order $4k$ is at least $|M(2k,k)|$. In particular we show that the number of magic squares of order $4k$ is at least $\frac{{2k \choose k}^2}{2}$.
We consider the properties of a slow-fast prey-predator system in time and space. We first argue that the simplicity of prey-predator system is apparent rather than real and there are still many of its hidden properties that have been poorly studied or overlooked altogether. We further focus on the case where, in the slow-fast system, the prey growth is affected by a weak Allee effect. We first consider this system in the non-spatial case and make its comprehensive study using a variety of mathematical techniques. In particular , we show that the interplay between the Allee effect and the existence of multiple timescales may lead to a regime shift where small-amplitude oscillations in the population abundances abruptly change to large-amplitude oscillations. We then consider the spatially explicit slow-fast prey-predator system and reveal the effect of different time scales on the pattern formation. We show that a decrease in the timescale ratio may lead to another regime shift where the spatiotemporal pattern becomes spatially correlated leading to large-amplitude oscillations in spatially average population densities and potential species extinction.
In the present paper, the symmetries admitted by semiconformal curvature tensor in semiconformally symmetric spacetime have been studied and we show that a four-dimensional spacetime admitting a proper semiconformal symmetry is semiconformally flat or of the Petrov type N. It is also shown that a four-dimensional spacetime with divergence-free semiconformal curvature tensor admitting a proper semiconformal symmetry is locally of the Petrov type O or has four distinct principal null directions. In both the cases, we found that if the spacetime admits an infinitesimal semiconformal Killing vector field then the scalar curvature of the spacetime vanishes.
The traveling wave and wave-train fronts correspond to the transition zones from a trivial steady state to a coexistence steady state and a limit cycle around the coexistence steady state, respectively. In this study, we establish the existence of both types of fronts for a reaction-diffusion system of prey-predator interactions with weak additive Allee effect in prey growth, Holling type II functional response and density dependent death rate for predators. For analytical simplicity, we consider immobile prey population. Under this consideration, the existence of both the traveling fronts are mathematically equivalent to the existence of point-to-point and point-to-cycle heteroclinic connections in R 3. The proof for the existence of traveling fronts relies on the construction of an unbounded wedged region, a shooting argument and an appropriate Lyapunov function. Also, by employing the Hopf bifurcation theorem we show the existence of traveling wave-train solution. In both the cases, we find successful invasion by the predator species. Further, we provide adequate numerical illustrations in order to corroborate our theoretical predictions. Our numerical simulations also suggest that the theoretically obtained minimum wave speed serves as an asymptotic speed of traveling wave propagation.
We study approximate Birkhoff–James orthogonality of bounded linear operators defined between normed linear spaces X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {X}$$\end{document} and Y.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Y}.$$\end{document} As an application of the results obtained, we characterize smoothness of a bounded linear operator T under the condition that K(X,Y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}(\mathbb {X},\mathbb {Y}),$$\end{document} the space of compact linear operators is an M-ideal in L(X,Y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {L}(\mathbb {X},\mathbb {Y}),$$\end{document} the space of bounded linear operators.
Recently, Lin introduced two new partition functions \(\hbox {PD}_{\mathrm{t}}(n)\) and \(\hbox {PDO}_{\mathrm{t}}(n)\), which count the total number of tagged parts over all partitions of n with designated summands and the total number of tagged parts over all partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for \(\hbox {PD}_{\mathrm{t}}(n)\) and \(\hbox {PDO}_{\mathrm{t}}(n)\), and conjectured some congruences modulo 8. In this paper, we prove the congruences modulo 8 conjectured by Lin and also find many new congruences and infinite families of congruences modulo some small powers of 2.
Making sense of transient dynamics
Ecological systems can switch between alternative dynamic states. For example, the species composition of the community can change or nutrient dynamics can shift, even if there is little or no change in underlying environmental conditions. Such switches can be abrupt or more gradual, and a growing number of studies examine the transient dynamics between one state and another—particularly in the context of anthropogenic global change. Hastings et al. review current knowledge of transient dynamics, showing that hitherto idiosyncratic and individual patterns can be classified into a coherent framework, with important general lessons and directions for future study.
Science , this issue p. eaat6412
In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let Δ k(n) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Wang and Yao, and Xia proved several infinite families of congruences modulo 7 for Δ 3(n) by using theta function identities. In this paper, we give a new proof of one result of Wang and Yao, and find three new infinite families of congruences modulo 7 for Δ 3(n). © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Non-local reaction-diffusion equation is an important area to study the dynamics of the individuals which compete for resources. In this paper, we describe a prey dependent predator-prey model with Holling type II functional response with a generalist predator. In particular, we want to see the behaviour of the system in the presence of nonlocal interaction. Introduction of nonlocal intra-specific competition in prey population leads to some new characteristics in comparison to the local model. Comparisons have been made between the local and nonlocal interactions of the system. The range of nonlocal interaction enlarges the parametric domain on which stationary patterns exist. The periodic oscillation for the local model in the Hopf domain can be stabilized by suitable limit of strong nonlocal interaction. An increase in the range of nonlocal interaction increases the Turing domain up to a certain level and then it decreases. Also, increasing the range of nonlocal interaction results in the overlap of nearby foraging areas and hence alters the size of the localized patches and formation of multiple stationary patches. Numerical simulations have been carried out to validate the analytical findings and to establish the existence of multiple stationary patterns, oscillatory solution, two-periodic solution and other spatio-temporal dynamics.
In this paper we study some geometric properties like parallelism, orthogonality and semi-rotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces $\mathbb{X} $ and $\mathbb{Y}$, assuming $\mathbb{X}$ to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces $\mathbb{X} $ and $\mathbb{Y}.$ We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff-James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semi-rotund points (semi-rotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semi-rotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semi-rotund space which is not strictly convex, if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.
We study Birkhoff-James orthogonality of bounded linear operators on complex Banach spaces and obtain a complete characterization of the same. By means of introducing new definitions, we illustrate that it is possible in the complex case, to develop a study of orthogonality of bounded (compact) linear operators, analogous to the real case. Furthermore, earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case, can be obtained as simple corollaries to our present study. In fact, we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case, in order to distinguish the complex case from the real case. We also study the left symmetric linear operators on complex two-dimensional $l_p$ spaces. We prove that $ T $ is a left symmetric linear operator on $ \ell_p^2{(\mathbb{C})}$ if and only if $ T $ is the zero operator.
The object of the present paper is to study a necessary condition
for the existence of weakly symmetric and weakly Ricci-symmetric LPSasakian manifolds admitting a quarter-symmetric metric connection.
Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led to some natural questions about the $p$-part of the classgroup of any CM Galois extension of $\Q$ as a module for $\Gal(K/Q)$, in the spirit of Herbrand-Ribet's theorem on the $p$-component of the class number of $Q(\zeta_p)$. In trying to formulate these questions, we are naturally led to consider $L(0,\rho)$, for $\rho$ an Artin representation, in situations where this is known to be nonzero and algebraic, and it is important for us to understand if this is $p$-integral for a prime $\p$ of the ring of algebraic integers $\bar{Z}$ in $C$, that we call {\it mod-$p$ Artin-Tate conjecture}. The most minor term in the class number formula, the number of roots of unity, plays an important role for us --- it being the only term in the denominator, is responsible for all poles!
In this paper, a type of Riemannian manifold (namely, pseudo semiconformally symmetric manifold) is introduced. Also the several geometric properties of such a manifold is investigated. Finally the existence of such a manifold is ensured by a proper example.
In a normed linear space X an element x is said to be orthogonal to another
element y in the sense of Birkhoff-James, written as $ x \perp_{B}y, $ iff $ \|
x \| \leq \| x + \lambda y \| $ for all scalars $ \lambda.$ We prove that a
normed linear space X is strictly convex iff for any two elements x, y of the
unit sphere $ S_X$, $ x \perp_{B}y $ implies $ \| x + \lambda y \| > 1~
\forall~ \lambda \neq 0. $ We apply this result to find a necessary and
sufficient condition for a Hamel basis to be a strongly orthonormal Hamel basis
in the sense of Birkhoff-James in a finite dimensional real strictly convex
space X. Applying the result we give an estimation for lower bounds of $ \|
tx+(1-t)y\|, t \in [0,1] $ and $ \| y + \lambda x \|, ~\forall ~\lambda $ for
all elements $ x,y \in S_X $ with $ x \perp_B y. $ We find a necessary and
sufficient condition for the existence of conjugate diameters through the
points $ e_1,e_2 \in ~S_X $ in a real strictly convex space of dimension 2. The
concept of generalized conjuagte diameters is then developed for a real
strictly convex smooth space of finite dimension.
The core object of this paper is to define and study a new class of analytic functions using the Ruscheweyh q-differential operator. We also investigate a number of useful properties of this class such structural formula and coefficient estimates for functions. We consider also the Fekete–Szegö problem in the class, we give some subordination results, and some other corollaries.
We present a new construction of Radon curves which only uses convexity methods. In other words, it does not rely on an auxiliary Euclidean background metric (as in the classical works of J. Radon, W. Blaschke, G. Birkhoff, and M. M. Day), and also it does not use typical methods from plane Minkowski Geometry (as proposed by H. Martini and K. J. Swanepoel). We also discuss some properties of normed planes whose unit circle is a Radon curve and give characterizations of Radon curves only in terms of Convex Geometry.
In this note we investigate the function Bk,ℓ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{k,\ell }(n)$$\end{document}, which counts the number of (k,ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k,\ell )$$\end{document}-regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for α≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 2$$\end{document} and n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document}, B3,113αn+5·3α-1-12≡0(mod11).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B_{3,11}\left( 3^{\alpha }n+\frac{5\cdot 3^{\alpha -1}-1}{2}\right) \equiv 0\ (\mathrm{mod\ }11). \end{aligned}$$\end{document}
The object of the present paper is to characterize a Riemannian manifold admitting a type of semi-symmetric non-metric connection.
We briefly review Ramanujan's theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a rich interplay between theta functions, eta-products and Eisenstein series. Transformation formulas of degrees five and seven for hypergeometric functions are obtained, and the paper ends with some series for 1/π similar to ones found by Ramanujan.
For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}}
{{z(q - 1)}} (z \in \mathbb{U}),$$ where $$\mathbb{U}$$ denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R
qλ
f is defined. Applying R
qλ
f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.
This paper deals with the study of Birkhoff--James orthogonality of a linear operator to a subspace of operators defined between arbitrary Banach spaces. In case the domain space is reflexive and the subspace is finite dimensional we obtain a complete characterization. For arbitrary Banach spaces, we obtain the same under some additional conditions. For an arbitrary Hilbert space H, we also study orthogonality to a subspace of the space of linear operators L(H), both with respect to operator norm as well as numerical radius norm.
Let L(X,Y) be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace Z⊂Y, we partially solve the operator version of Birkhoff–James orthogonality problem, if T∈L(X,Y) is orthogonal to L(X,Z), when does there exist a unit vector x0 such that ||T(x0)||=||T|| and T(x0) is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for d(T,L(X,Z)) in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for Z⊂Y is a L¹-predual space as well as a M-ideal in Y, we show that if T∈K(X,Y) is orthogonal to L(X,Z), then there is a unit vector x0 with ||T(x0)||=||T|| and T(x0) is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved.
International audience
The first part of this paper is a survey on Ramachandra's contribution to transcendental number theory included in his 1968 paper in Acta Arithmetica. It includes a discussion of pseudo-algebraic points of algebraically additive functions. The second part deals with applications to density statements related with a conjecture due to B.~Mazur. The next part is a survey of other contributions of Ramachandra to transcendence questions (on the numbers $2^{\pi^k}$, a note on Baker's method, an easy transcendence measure for $e$). Finally, related open questions are raised.
We characterize Birkhoff-James orthogonality of continuous vector-valued functions on a compact topological space. As an application of our investigation, Birkhoff-James orthogonality of real bilinear forms are studied. This allows us to present an elementary proof of the well-known Bhatia-Šemrl Theorem in the real case.
Radon planes are two-dimensional real normed spaces in which Birkhoff(-James) orthogonality is symmetic. It is shown that every Radon plane is isometrically isomorphic to a special Day–James space generated by a pair of an absolute norm and its dual norm.
We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in normed linear spaces. In this context, the key aspect of our study is to consider norming sequences for a bounded linear operator, instead of norm attaining elements. We also obtain a complete characterization of smoothness of bounded linear operators between real normed linear spaces, when the corresponding norm attainment set non-empty. This illustrates the importance of the norm attainment set in the study of smoothness of bounded linear operators. Finally, we prove that Gâteaux differentiability and Fréchet differentiability are equivalent for compact operators in the space of bounded linear operators between a reflexive Kadets-Klee Banach space and a Fréchet differentiable normed linear space, endowed with the usual operator norm.
Although the spatial dimension of ecosystem dynamics is now widely recognized, the specific mechanisms behind species patterning in space are still poorly understood and the corresponding theoretical framework is underdeveloped. Going beyond the classical Turing scenario of pattern formation, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation illustrates how mathematical modeling and numerical simulations can lead to greater understanding of these issues. It takes a unified approach to population dynamics and epidemiology by presenting several ecoepidemiological models where both the basic interspecies interactions of population dynamics and the impact of an infectious disease are explicitly considered. The book first describes relevant phenomena in ecology and epidemiology, provides examples of pattern formation in natural systems, and summarizes existing modeling approaches. The authors then explore nonspatial models of population dynamics and epidemiology. They present the main scenarios of spatial and spatiotemporal pattern formation in deterministic models of population dynamics. The book also addresses the interaction between deterministic and stochastic processes in ecosystem and epidemic dynamics, discusses the corresponding modeling approaches, and examines how noise and stochasticity affect pattern formation. Reviewing the significant progress made in understanding spatiotemporal patterning in ecological and epidemiological systems, this resource shows that mathematical modeling and numerical simulations are effective tools in the study of population ecology and epidemiology.
In this paper, we study Birkhoff-James orthogonality of bounded linear operators and give a complete characterization of Birkhoff-James orthogonality of bounded linear operators on infinite dimensional real normed linear spaces. As an application of the results obtained, we prove a simple but useful characterization of Birkhoff-James orthogonality of bounded linear functionals defined on a real normed linear space, provided the dual space is strictly convex. We also provide separate necessary and sufficient conditions for smoothness of bounded linear operators on infinite dimensional normed linear spaces.
We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on an analysis of the expected characteristic polynomial of a union of random perfect matchings and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real-rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions introduced recently by the authors. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
The object of the present paper is to study weakly semiconformally symmetric manifolds (WSCS)n. At first some geometric properties of (WSCS)n (n > 2) have been studied. Finally, we consider the decomposability of (WSCS)n.
In this paper, we introduce a class of normalized analytic functions defined in
the open unit disc. We study the unified class of functions with bounded Mocanu
variation which map the open unit disk onto conic regions using q -derivative. We
invistigate several Interesting mapping properties, certain inclusion results and
generalized type of q -Bernardi integral operator for this class.
The main object of the present paper is to define q-analogue of Ruscheweyh operator for multivalent functions. We investigate a number of useful properties including coefficient estimates, sufficiency criteria and the familiar Feke–Szegö type inequality for a newly defined class. Several known consequences of the main results are also pointed out.
Working mostly in isolation, Ramanujan noted striking and sometimes still unproved results in series, special functions and number theory.
For Banach spaces X, Y, in the space of bounded linear operators , we examine the relation between being a smooth point versus being a smooth point. Motivated by some results in the recent paper of Paul et al., we give some sufficient conditions for the validity of such a statement for spaces of operators.
Let $B_{k,\ell }(n)$ denote the number of $(k,\ell )$ -regular bipartitions of $n$ . Employing both the theory of modular forms and some elementary methods, we systematically study the arithmetic properties of $B_{3,\ell }(n)$ and $B_{5,\ell }(n)$ . In particular, we confirm all the conjectures proposed by Dou [‘Congruences for (3,11)-regular bipartitions modulo 11’, Ramanujan J.40 , 535–540].
We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let , where is a real Banach space and is a real normed linear space. We find sufficient condition for for some with , and use it to show that T is a smooth point in if T attains its norm at unique (upto multiplication by scalar) vector , Tx is a smooth point of and for all closed subsets C of with . For operators on a Hilbert space we show that for some with if and only if the norm attaining set for some finite dimensional subspace and . We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.
In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff-James orthogonality techniques, we give a necessary condition for a bounded linear operator attaining norm at a particular point of the unit sphere. We prove a number of corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff-James orthogonality. Restricting our attention to $ l_{p}^{2} (p \in \mathbb{N}\setminus \{ 1 \})$ spaces, we obtain an upper bound for the number of points at which any linear operator, which is not a scalar multiple of an isometry, may attain norm.
Let (Formula presented.) denote the number of (Formula presented.)-regular bipartitions of (Formula presented.). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function (Formula presented.). In particular, we shall prove an infinite family of congruences: for (Formula presented.) and (Formula presented.), (Formula presented.)In addition, we will also give an alternative proof of one infinite family of congruences for (Formula presented.), the number of (Formula presented.) regular partitions of (Formula presented.), due to Webb.
In this paper, we give an elementary proof of Ramanujan's Eisenstein series of level 7. In the process, we also prove four Eisenstein series of level 14 due to S. Cooper and D. Ye [4].
In this paper, we study the function Bℓ(n) which counts the number of ℓ-regular bipartitions of n. Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). In particular, using Ramanujan’s two modular equations of degree 7, we prove an infinite family of congruences: for α≥2 and n≥0, $$B_7 \biggl(3^{\alpha}n+\frac{5\cdot 3^{\alpha-1}-1}{2} \biggr)\equiv 0\ ({ \rm mod\ }3). $$ In addition, we give an elementary proof of two infinite families of congruences modulo 3 satisfied by the 7-regular partition function due to Furcy and Penniston (Ramanujan J. 27:101–108, 2012). We also present two conjectures for B13(n) modulo 3.