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

Arithmetic springer theorem and nn-universality under field extensions

Based on BONGs theory, we prove the norm principle for integral and relative integral spinor norms of quadratic forms over general dyadic local fields, respectively. By virtue of these results, we further establish the arithmetic version of Springer's theorem for indefinite quadratic forms. Moreover, we solve the lifting problems on $n$-universality over arbitrary local fields

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