NOTES ON GLAISHER''''S CONGRUENCES

Let p be an odd prime and let n ≥ 1,k ≥ 0 and r be integers. Denote by Bk the kth Bernoulli number. It is proved that (i) If r≥1 is odd and suppose p≥r+4, thenp-1∑j=11/(np+j)=-(2n+1+(x+1)/2(x+2)Bp-r-2p2(modp3).(ii)IFr≥2is even and suppose p≥r+3,thenp-1∑j=11/(np+j)+=r/r+1Bp-x-1p(modp2).(iii)p-1∑j=11/(np+j)p-2=-(2n+1)p(modp2).Thisesult generalizes the Glaisher's congruence. As a corollary, a generalization of the Wolstenholme's theorem is obtained.     

Binomial coefficients, Catalan numbers and Lucas quotients

Let p be an odd prime and let a,m ∈ Z with a 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0kpa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.     

Congruences involving generalized central trinomial coefficients

For integers b and c the generalized central trinomial coefficient Tn(b,c)denotes the coefficient of xnin the expansion of(x2+bx+c)n.Those Tn=Tn(1,1)(n=0,1,2,...)are the usual central trinomial coefficients,and Tn(3,2)coincides with the Delannoy number Dn=n k=0n k n+k k in combinatorics.We investigate congruences involving generalized central trinomial coefficients systematically.Here are some typical results:For each n=1,2,3,...,we have n-1k=0(2k+1)Tk(b,c)2(b2-4c)n-1-k≡0(mod n2)and in particular n2|n-1k=0(2k+1)D2k;if p is an odd prime then p-1k=0T2k≡-1p(mod p)and p-1k=0D2k≡2p(mod p),where(-)denotes the Legendre symbol.We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.     

A specific family of cyclotomic polynomials of order three

Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x).We say Φn(x) is flat if A(n) = 1.In this paper,for odd primes p q r and 2r ≡ 1(mod pq),we prove that Φpqr(x) is flat if and only if p = 3 and q ≡ 1(mod 3).     

Maximum Modulus of Polynomials

Let P(z) =n∑j=0 a_jz~j be a polynomial of degree n and let M(P, r) = max|z|=r|P(z)|. If P(z) ≠ 0 in |z| 1, then M( P, r) ≥ ((1 + r)/ (1 + ρ))~ n M( P, ρ).The result is best possible. In this paper we shall present a refinement of this result and some other related results.     

A nontrivial product in the stable homotopy groups of spheres

Let A be the mod p Steenrod algebra for p an arbitrary odd prime. In 1962, Li-uleviciusdescribed hi and bk in Ext (A|*,*) (Zp, Zp) having bigrading (1,2pi(p-1))and (2,2pk+1 x(p - 1)), respectively. In this paper we prove that for p ≥ 7,n ≥ 4 and 3 ≤ s < p - 1, (Zp,Zp) survives to E∞ in the Adams spectral sequence, where q = 2(p - 1).     

ON THE DIOPHANTINE EQUATION X4-Dy2=1(II)

For the Diophantine equation x^4 — Dy^2 = 1 (1) where D>0 and is not a perfect square, we prove the following theorems in this paper. Theorem 1. If D\[{\not \equiv }\]7 (mod 8)，D=p1p2...ps，s≥2，where pi（i = 1，…，s) are distincyt primes，p1≡1(mod 4) such that either 2p1=a^2+b^2，а≡\[ \pm \]3(mod 8)，b三\[ \pm \]3(mod 8) or there is a j(2≤j≤s), for which Legendre symbal \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\],and pi≡7(mod8) (i=2,..., s) or pi≡3(mod 8) (i=2,..., s), then (1) has no solutions in positive integer x,y. Theorem 2. If D=p1...ps,s≥2, where pi（i = 1，…，s) are distinct primes, and pi≡3(mod 4)（i = 1，…，s), then (1) has no solutions in positive integer x, y. Theorem 3. The equation (1) with D=2p1...ps has no solutions in positive integer x, y, if (1) p1≡(mod 4), pi≡7(mod 8) (i = 2, ???, s), snch that either 2p1 = a^2+b^2 a≡\[ \pm \]3(mod 8)，b≡\[ \pm \]3(mod 8)or there is a j (2≤j≤s)，for which \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\]; or (2) p1≡5(mod8),pi≡3(mod8) (i = 2,..., s)； or ⑶p1≡5(mod8)，pi≡7(mod 8) (i=2，…，s). Corollary of theorem 3. If D = 2pq, p≡5(mod 8), q≡3(mod 4), where p, q are distinct primes, then (1) has no solutions in positive integer x, y. Theorem 4. If D=2p1...ps, pi≡3(mod 4)(0 = 1,...,s), then (1) has no solutions In positive integer x, y.     

RESEARCH ANNOUNCEMENTS On Generalized D.H.Lenmer''''s Problem

If q is an odd integer, q≥3,for any integers α, (α,q) = 1,there exsits a positiveinteger α, so tbat αα≡1(mod q) and 1≤α≤q - 1. Let L(q) = {α|α∈Z,1≤α≤q - 1, (α,q) = 1 and α +α=1(mod 2)}. (1)About the property of elements of L(q) is a generalization of a problem of D. H.Lenmer ([1],p. 12). In [3], it was conjectured that     

The automorphism group of a generalized extraspecial p-group

In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .     

Existence of Some Specific Elements in Finite Fields北大核心CSCD

Let q = pk and Fqn be the extension field of Fq of degree n, where p is an odd prime and n, k are positive integers. The main contribution of this paper is as follows: If n | (q − 1), k ≥ 11, n ≥ 14 or n (q − 1), k ≥ 10, n ≥ 8, then there exists a primitive element α in Fqn such that α + α−1 is a normal element, and 1 + α2 is a square element, and there exists a normal element β, such that β + β−1 is a primitive element, and 1 + β2 is a square element. © 2022 Chinese Academy of Sciences. All rights reserved.     

On an exponential sum

Let p be a prime number, n be a positive integer, and ƒ(x) = ax^k + bx. We put

Semiclassical symmetric Schrödinger equations: Existence of solutions concentrating simultaneously on several spheres

We study the radially symmetric Schr?dinger equation

The Cauchy-Dirichlet problem for the heat equation in Besov spaces

The solvability in anisotropic spaces , σ ∈ ℝ₊, p, q ∈ (1, ∞), of the heat equation u_t − Δu = f in Ω^T ≡ (0, T) × Ω is studied under the boundary and initial conditions u = g on S^T, u|_t=0 = u₀ in Ω, where S is the boundary of a bounded domain Ω ⊂ ℝⁿ. The existence of a unique solution of the above problem is proved under the assumptions that and under some additional conditions on the data. The existence is proved by the technique of regularizers. For this purpose the local-in-space solvability near the boundary and near an interior point of Ω is needed. To show the local-in-space existence, the definition of Besov spaces by the dyadic decomposition of a partition of unity is used. This enables us to get an appropriate estimate in a new and promising way without applying either the potential technique or the resolvent estimates or the interpolation. Bibliography: 26 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 40–97.     

On the riemann zeta-function and the divisor problem

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If with , then we obtain

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p (\mathbb{S}^{{d - 1}} ) (0 < p ≤ 1)

Let be a unit sphere of the d–dimensional Euclidean space ℝ ^d and let (0 < p ≤ 1) denote the real Hardy space on For 0 < p ≤ 1 and let E _j (f,H ^p ) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space Given a distribution f on its Cesàro mean of order δ > –1 is denoted by For 0 < p ≤ 1, it is known that is the critical index for the uniform summability of in the metric H ^p . In this paper, the following result is proved: Theorem Let 0<p<1 and Then for

On the decay rate of (p, A)-lacunary series

A power series with radius of convergence equal 1 is called a (p,A)-lacunary one if n_k ≥ Ak^p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition

The Law of Iterated Logarithm of Rescaled Range Statistics for AR（1） Model

Let {Xn,n ≥ 0} be an AR（1） process. Let Q（n） be the rescaled range statistic, or the R/S statistic for {Xn} which is given by （max1≤k≤n（∑j=1^k（Xj - ^-Xn）） - min 1≤k≤n（∑j=1^k（ Xj - ^Xn ））） /（n ^-1∑j=1^n（Xj -^-Xn）^2）^1/2 where ^-Xn = n^-1 ∑j=1^nXj. In this paper we show a law of iterated logarithm for rescaled range statistics Q（n） for AR（1） model.     

Hua＇s Theorem on Five Almost Equal Prime Squares

In this paper we sharpen Hua＇s result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N=p1^2＋p2^2＋p3^2＋p4^2＋p5^2,with │pj-√N/5│≤U=N^1/2-1/28＋ε,where pj are primes.     

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.