In this paper, we show that the title equation, where Fm is the mth Fibonacci number, in positive integers (k,n,p,q) with k>1 entails max⁡{k,n,p,q}≤102500.

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if G is such a graph, then G2 is (Δ(G) + 72)-degenerate. This implies an upper bound of Δ(G) ∣ 73 on the chromatic number of G2 as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon-Tarsi number, and correspondence chromatic number. We also show

Dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if P has dimension d, then to know whether x ≤ y in P it is enough to check whether x ≤ y in each of the d linear extensions of a witnessing realizer. Focusing on the encoding aspect, Nešetřil and Pudlák defined a more expressive version of dimension. A poset P has Boolean dimension

We prove the following statement. Let f ∈ ℝ[x1,…,xd], for some d ≥ 3, and assume that f depends non-trivially in each of x1,…, xd. Then one of the following holds. (i) For every finite sets A1,…, Ad ⊂ℝ, each of size n, we have $$\left| {f\left( {{A_1} \times \ldots \times {A_d}} \right)} \right| = \Omega \left( {{n^{3/2}}} \right),$$ with constant of proportionality that depends on deg f. (ii) f is

Let G = (V, E) be a finite graph. For v ∈ V we denote by Gv the subgraph of G that is induced by v’s neighbor set. We say that G is (a,b)-regular for a>b> 0 integers, if G is a-regular and Gv is b-regular for every v ∈ V. Recent advances in PCP theory call for the construction of infinitely many (a,b)-regular expander graphs G that are expanders also locally. Namely, all the graphs {Gv ∣ v ∈ V} should

In this paper we extend the result of [5] to the quantum version. We present 11 examples of f's and discuss f-deformed quantum mechanics in one dimension for each f which possesses the f-deformed translational symmetry.

Formula for the size of the scatterer is derived explicitly in terms of the scattering amplitude corresponding to this scatterer. By the scatterer either a bounded obstacle D or the support of the compactly supported potential is meant.

Controlled time-decaying harmonic potentials decelerate the velocity of charged particles; however, the particles are never trapped by the harmonic potentials. This physical phenomenon changes the threshold between the short-range and long-range classes of the potential through physical wave operators. We herein report that the threshold is 1/(1 − λ) for some 0 ≤ λ < 1/2, as determined using the mass

We introduce a class of novel ℤ2 × ℤ2-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the ℤ2 × ℤ2-graded color superalgebras is presented. It turns out that infinitely many members

Theoretical study of superluminal sources of electromagnetic radiation boosted after the discovery of Cherenkov–Vavilov radiation. Later, the way to create fictitious sources moving superluminally was suggested. Different approaches have been proposed for the research of the distribution of the potential and the fields radiated by the superluminally moving charges. The simplest idealized cases of uniform

It is well known that Galilean conformal algebras play important roles in the nonrelativistic anti-de Sitter/conformal field theory correspondence. The finite Lie conformal algebras PG(a,b) of planar Galilean type can be viewed as Lie conformal analogues of certain planar Galilean conformal algebras. In this paper, we classify finite irreducible conformal modules over PG(a,b) for all complex numbers

The geometric framework of Double Field Theory (DFT) can be constructed on a para-Hermitian manifold. The canonical, generalized-metric connections and the global expression of the corresponding covariant derivative, a generalization of the kinematical structure of DFT, generalized curvature, a corresponding generalized Lie derivative for the Leibniz algebroid on the tangent bundle are constructed

We derive the Simon–Wolff localization criterion from the spectral averaging using an intuitive measure-theoretic lemma.

We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding a ring of smooth functions into the category of loci. This permits us to define a category of functorial differential spaces. By suitably choosing various algebras

Suppose ρ1,ρ2 are two ℓ-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by some representation, it is given by a pair

As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges

In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions of our model with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and

Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A⊆Fp with p prime and |A|<0.0045p, (i) if |A+A|<2.59|A|−3 and |A|>100, then A is contained in an arithmetic progression of size |A+A|−|A|+1, and (ii) if |A−A|<2.6|A|−3, then A is contained in an arithmetic progression of size |A−A|−|A|+1. The improvement comes from using the properties of higher energies

We show that an additive Hilbert cube (in prime fields) of sufficiently large dimension always meets certain kinds of arithmetic sets, namely, product sets and reciprocal sets of sumsets satisfying certain technical conditions.

For positive integers α and β, we define an (α,β)-walk to be any sequence of positive integers satisfying wk+2=αwk+1+βwk. We say that an (α,β)-walk is n-slow if ws=n with s as large as possible. Slow (1,1)-walks have been investigated by several authors. In this paper we consider (α,β)-walks for arbitrary positive α,β. We derive a characterization theorem for these walks, and with this we prove several

Values of quaternionic modular forms are related to twisted central L-values via periods and a theorem of Waldspurger. In particular, certain twisted L-values must be non-vanishing for forms with no zeroes. Here we study, theoretically and computationally, zeroes of definite quaternionic modular forms of trivial weight. Local sign conditions force certain forms to have trivial zeroes, but we conjecture

We study solvability of the Diophantine equationn2n=∑i=1kai2ai, in integers n,k,a1,…,ak satisfying the conditions k≥2 and ai

We prove that q+1-regular Morgenstern Ramanujan graphs Xq,g (depending on g∈Fq[t]) have diameter at most (43+ε)logq⁡|Xq,g|+Oε(1) (at least for odd q and irreducible g) provided that a twisted Linnik–Selberg conjecture over Fq(t) is true. This would break the 30 year-old upper bound of 2logq⁡|Xq,g|+O(1), a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by

We estimate the growth of cuspidal cohomology of GL2(AQ). Quantitatively, we provide bounds on the total number of normalised eigenforms of Hecke operators which are obtained by automorphic induction from Hecke characters of imaginary quadratic fields grows as level structure varies. We further investigate how much of cuspidal cohomology of GL3(AQ) is obtained by symmetric square transfer from GL2(AQ)

Let F be a non-archimedean locally compact field. We study a class of Langlands-Shahidi pairs (H,L), consisting of a quasi-split connected reductive group H over F and a Levi subgroup L which is closely related to a product of restriction of scalars of GL1's or GL2's. We prove the compatibility of the resulting local factors with the Langlands correspondence. In particular, let E be a cubic separable

We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical

Two number fields with equal Dedekind zeta function are not necessarily isomorphic. However, if the number fields have equal sets of Dirichlet L-series then they are isomorphic. We extend this result by showing that the isomorphisms between the number fields are in bijection with L-series preserving isomorphisms between the character groups.

The minus space Mk!−(p) is defined to be the subspace of the space Mk!(p) of weakly holomorphic weight k modular forms for Γ0(p) consisting of all eigenforms of the Fricke involution Wp with eigenvalue −1. In this paper, we study congruences for Hecke eigenvalues in minus spaces. This is extended results for Choi and Kim (2011) [CK11]. We also find congruence relations satisfied by the Fourier coefficients

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer n is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be σ(n)/2. Generalizing even further, we call n a k-layered number if its divisors can be partitioned into k sets with equal sum. In this paper, we completely

Let K be a number field and S be a finite set of valuations of K containing the archimedean valuations. Let O be the ring of S-integers. For A∈SL2(O) and k≥1, we define matrix-factorization varieties Vk(A) over O which parametrize factoring A into a product of k elementary matrices beginning with lower triangular; the equations defining Vk(A) are written in terms of Euler's continuant polynomials.

A non-singular complete irreducible algebraic curve Fk,n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k,n), where n,k≥2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H≅Zkn of automorphisms such that Fk,n/H is isomorphic to PK1 and it has exactly (n+1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse

Let χ be a nonprincipal Dirichlet character modulo a prime number p⩾3 of order k⩾2. DefineAp(χ):=1p−1∑1⩽N⩽p−1∑1⩽n1,n2⩽Nχ(n1)=χ(n2)1. We prove thatAp(χ)=p(2p−1)6k+(k−1)(p+1)12k+aχp2π2k(p−1)∑j=1k/2|L(1,χ2j−1)|2 where aχ:=(1−χ(−1))/2.

Text Let G be a finite abelian group and S be a sequence with elements of G. Let Σ(S)⊂G denote the set of group elements which can be expressed as a sum of a nonempty subsequence of S. We call S zero-sum free if 0∉Σ(S). In this paper, we study |Σ(S)| when S is a zero-sum free sequence of elements from G and 〈S〉 is not cyclic. We improve the results of A. Pixton and P. Yuan on this topic. In particular

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫n2 log−3n, where a hinge is identified by fixing two pairwise distances in a point triple. This is achieved via strengthening (modulo a logn factor) of the Guth- Katz estimate for the number of pairwise intersections of lines in ℝ3, arising in the context of the plane Erdős distinct distance

We show that a quasirandom k-uniform hypergraph G has a tight Euler tour subject to the necessary condition that k divides all vertex degrees. The case when G is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the k-subsets of an n-set.

Let X:y2=f(x) be a hyperelliptic curve over Q(T) of genus g≥1. Assume that the jacobian of X over Q(T) has no subvariety defined over Q. Denote by Xt the specialization of X to an integer T=t, let aXt(p) be its trace of Frobenius, and let AX,r(p)=1p∑t=1paXt(p)r be its r-th moment. The first moment is related to the rank of the jacobian JX(Q(T)) by a generalization of a conjecture of Nagao:limX→∞⁡1X∑p≤X−AX

We prove that for any associator, two specific families of coefficients of the associator can be expressed in terms of coefficients of lower depth. Combining these results to our notions of adjoint p-adic multiple zeta values and multiple harmonic values, we obtain a new point of view on the question of relating p-adic and finite multiple zeta values, and a few other application to the study of p-adic

We describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta

We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for n>1 we prove thatτ(n)∉{±1,±3,±5,±7,±691}. This result is an example of general theorems (see Theorems 1.2 and 1.3 of [2]) for newforms with trivial mod 2 residual Galois representation. Ramanujan's well-known congruences for τ(n) allow for the simplified proof in these special cases

Let X be a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Suppose G is a finite group acting faithfully on X such that G has non-trivial cyclic Sylow p-subgroups. We show that the decomposition of the space of holomorphic differentials of X into a direct sum of indecomposable k[G]-modules is uniquely determined by the lower ramification groups

We present the basics of the local theory, which arises from global Rankin-Selberg integrals, attached to pairs of irreducible globally generic cuspidal automorphic representations of the quasi-split unitary group U2n+1 and ResE/FGLm, for a quadratic extension of number fields E/F, when m>n.

The authors have recently obtained a lower bound of the Hausdorff dimension for the sets of vectors (x1,…,xd)∈[0,1)d with large Weyl sums, namely of vectors for which|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|⩾Nα for infinitely many integers N⩾1. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.

Ideal class pairings map the rational points of rank r≥1 elliptic curves E/Q to the ideal class groups CL(−D) of certain imaginary quadratic fields. These pairings imply thath(−D)≥12(c(E)−ε)(log⁡D)r2 for sufficiently large discriminants −D in certain families, where c(E) is a natural constant. These bounds are effective, and they offer improvements to known lower bounds for many discriminants.

In this paper we consider iterated integral representations of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biprojective hypersurfaces of bidegree (1,2) in sufficiently many variables. This confirms the Manin conjecture for this variety.

We use Maeda's Conjecture to prove that the Rankin-Cohen bracket of an eigenform and any modular form is only an eigenform when forced to be because of the dimensions of the underlying spaces. This occurs, for example, when the Rankin-Cohen bracket covers the entirety of Sn. We further determine when the Rankin-Cohen bracket of an eigenform and modular form is not forced to produce an eigenform and

In this paper we suggest a way to understand the structure of depth-graded motivic Lie subalgebra generated by the depth one part for the neutral Tannakian category mixed Tate motives over Z. We will show that from an isomorphism conjecture proposed by K. Tasaka we can deduce the F. Brown's matrix conjecture and the nondegeneracy conjecture about depth-graded motivic Lie subalgebra generated by the

We complete the proof of a Siegel type statement for finitely generated Φ-submodules of Ga under the action of a Drinfeld module Φ.

The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic

We prove various converse theorems for automorphic forms on Γ0(N), each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N∈{18,20,24} - these integers are the smallest multiples of 4 or 9 not covered by earlier work of Conrey–Farmer. This development is a consequence of finding generating sets for Γ0(N)

We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle α with continued fraction expansion [0;a1,a2,…] is either tqk with 1≤t≤ak+1 (a multiple of a denominator qk of a convergent of α) or qk,ℓ (a denominator qk,ℓ of a semiconvergent of α)

In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group G is an almost simple group in which the socle A has prime index p. Each Hopf-Galois structure is associated to a group N of the same order as G. We shall give necessary criteria on these N in terms of their group-theoretic properties, and determine the number of Hopf-Galois structures associated to A×Cp

We prove that the equation (x−3r)3+(x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3+(x+3r)3=yp only has solutions which satisfy xy=0 for 1≤r≤106 and p≥5 prime. This article complements the work on the equations (x−r)3+x3+(x+r)3=yp in [2] and (x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3=yp in [1]. The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution

For a prime number p and an integer n we determine the Galois cohomology groups of the class group of the normal closure of Q(np) to a certain extent and use this information to prove a result about the group structure of the class group.

There are a lot of effective, ineffective and explicit results concerning power values and polynomial values of binomial coefficients. Also, many papers deal with generalizations of these problems, involving polygonal numbers and pyramidal numbers. In this paper we prove effective and ineffective theorems concerning polynomial values of figurate numbers. Our results yield common extensions and generalizations

In this paper it is shown that there is an absolute constant κ>0 with the following property. For any prime p and nonempty subsets A,B of Z/pZ such that 1<|A|

For a real parameter r, the RSA integers are integers which can be written as the product of two primes pq with p

An integer of the form Tx=x(x+1)2 for some positive integer x is called a triangular number. A ternary triangular form aTx+bTy+cTz for positive integers a,b and c is called regular if it represents every positive integer that is locally represented. In this article, we prove that there are exactly 49 primitive regular ternary triangular forms.

Let p=(P) be any prime of Fq[t], let m be any ideal of Fq[t] not divisible by p and consider the space of Drinfeld cusp forms of level mp, i.e. for the modular group Γ0(mp). Using degeneracy maps, traces and Fricke involutions we offer definitions for p-oldforms and p-newforms which turn out to be subspaces stable with respect to the action of the Atkin operator UP. We provide eigenvalues and/or slopes

We give congruences modulo powers of p∈{3,5,7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and the first author and continuing work done by the authors and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at ∞.