Some Implications of the Gessel Identity

We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.

关于整数表示的新猜想(Ⅰ)（英文）

本文研究把自然数写成方幂的加权和表示问题并提出这方面的一些新猜想.例如:我们证明本质上只有9个形如aw^h+bx^i+cy^j+dz^k(其中a,b,c,d为正整数,h,i,j,k∈{2,3,4…,}且h,i,j,k中至少有一个为2)的多项式使得每个非负整数n可表成aw^h+bx^i+cy^j+dz^k的形式(其中w,x,y,z为非负整数).我们的一个猜想断言如果f(w,x,y,z)是9个多项式w^2+x^3+2y^3+cz^3(c=3,4,5,6),w^2+x^3+2y^3+dz^4(d=1,3,6),2w^2+x^3+4y^3+z^4,w^2+x^3+y^4+2z^4之一那么就有{f(w,x,y,z):w,x,y,z=0,1,2,…,}={0,1,2,…}.我们也猜测大于1的整数n可表成x^4+y^3+z^2+2~k的形式,其中x,y,z为非负整数且k为正整数.

幂商和与幂和差不等式及其应用

本文根据加权均值不等式推广了文[1]的主要结论,并利用推广结构对一组国内外不等式赛题进行了推广简证。