On the sum of reciprocal generalized Fibonacci numbers

In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers

On n-ADC integral quadratic lattices over totally real number fields

In the paper, we extend the ADC property to the representation of quadratic lattices by quadratic lattices, which we define as $n$-ADC-ness. We explore the relationship between $n$-ADC-ness, $n$-regularity and $n$-universality for integral quadratic lattices. Also, for $n\ge 2$, we give necessary and sufficient conditions for an integral quadratic lattice with rank $n+1$ or $n+2$ over local fields to be $n$-ADC. Moreover, we show that over any totally real number field $F$, a positive definite integral $\mathcal{O}_{F}$-lattice with rank $n+1$ is $n$-ADC if and only if it is $\mathcal{O}_{F}$-maximal of class number one

A conjecture on the primitive degree of Tensors

In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This confirms a conjecture of Shao [7].Comment: 8 page

On kk-universal quadratic lattices over unramified dyadic local fields

Let $k$ be a positive integer and let $F$ be a finite unramified extension of $\mathbb{Q}_2$ with ring of integers $\mathcal{O}_F$. An integral (resp. classic) quadratic form over $\mathcal{O}_F$ is called $k$-universal (resp. classically $k$-universal) if it represents all integral (resp. classic) quadratic forms of dimension $k$. In this paper, we provide a complete classification of $k$-universal and classically $k$-universal quadratic forms over $\mathcal{O}_F$. The results are stated in terms of the fundamental invariants associated to Jordan splittings of quadratic lattices.Comment: 40 page

On nn-universal quadratic forms over dyadic local fields

Let $n \ge 2$ be an integer. We give necessary and sufficient conditions for an integral quadratic form over dyadic local fields to be $n$-universal by using invariants from Beli's theory of bases of norm generators. Also, we provide a minimal set for testing $n$-universal quadratic forms over dyadic local fields, as an analogue of Bhargava and Hanke's 290-theorem (or Conway and Schneeberger's 15-theorem) on universal quadratic forms with integer coefficients

The mass of shifted lattices and class numbers of inhomogeneous quadratic polynomials

In this paper, we investigate class numbers of shifted quadratic lattices $L+\frac{\boldsymbol{u}}{c}$ with $\boldsymbol{u}\in L$ and odd conductor $c\in \mathbb{N}$. For a lattice $L$ whose genus only contains one class, we determine a lower bound for the number of classes in the genus of $L+\frac{\boldsymbol{u}}{c}$ depending on $c$. As a result, we obtain an explicit bound $c_0$ such that any such shifted lattice with one class in its genus must have conductor smaller than $c_0$, restricting the possible choices of such $L+\frac{\boldsymbol{u}}{c}$ to a finite set