On the Lower Markov Spectrum.
Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and for n ≥ 1, and V₀ = 2, V₁ = P and for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and . We show that there is no integer x such that when m ≥ 1 and r is an even integer. Also we completely solve the equation for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and Q ≡ 1 (mod...
We prove the last of five outstanding conjectures made by R. M. Robinson from 1965 concerning small cyclotomic integers. In particular, given any cyclotomic integer β all of whose conjugates have absolute value at most 5, we prove that the largest such conjugate has absolute value of one of four explicit types given by two infinite classes and two exceptional cases. We also extend this result by showing that with the addition of one form, the conjecture is true for β with magnitudes up to 5 + 1/25....
Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) with d ∈ N for nonsingular , i=1,...,n.
We prove that the sums of independent random vectors satisfy , t ≥ 0.
We obtain a metrical property on the asymptotic behaviour of the maximal run-length function in the Lüroth expansion. We also determine the Hausdorff dimension of a class of exceptional sets of points whose maximal run-length function has sub-linear growth rate.
For the cyclotomic -extension of an imaginary quadratic field , we consider the Galois group of the maximal unramified pro--extension over . In this paper, we give some families of for which is a metabelian pro--group with the explicit presentation, and determine the case that becomes a nonabelian metacyclic pro--group. We also calculate Iwasawa theoretically the Galois groups of -class field towers of certain cyclotomic -extensions.