1,112 research outputs found
Coincidences in generalized Lucas sequences
For an integer , let be the generalized
Lucas sequence which starts with ( terms) and each term
afterwards is the sum of the preceding terms. In this paper, we find all
the integers that appear in different generalized Lucas sequences; i.e., we
study the Diophantine equation in nonnegative integers
with . The proof of our main theorem uses lower
bounds for linear forms in logarithms of algebraic numbers and a version of the
Baker-Davenport reduction method. This paper is a continuation of the earlier
work [4].Comment: 14 page
Fibonacci-Lucas SIC-POVMs
We present a conjectured family of SIC-POVMs which have an additional
symmetry group whose size is growing with the dimension. The symmetry group is
related to Fibonacci numbers, while the dimension is related to Lucas numbers.
The conjecture is supported by exact solutions for dimensions
d=4,8,19,48,124,323, as well as a numerical solution for dimension d=844.Comment: The fiducial vectors can be obtained from
http://sicpovm.markus-grassl.de as well as from the source files. v2:
precision for the numerical solution in dimension 844 increased to 150 digits
and new exact solution for dimension 323 adde
- β¦