On Taylor's conjecture for Kummer orders
On Terai's conjecture concerning Pythagorean numbers
On terms of linear recurrence sequences with only one distinct block of digits
In 2000, Florian Luca proved that F₁₀ = 55 and L₅ = 11 are the largest numbers with only one distinct digit in the Fibonacci and Lucas sequences, respectively. In this paper, we find terms of a linear recurrence sequence with only one block of digits in its expansion in base g ≥ 2. As an application, we generalize Luca's result by finding the Fibonacci and Lucas numbers with only one distinct block of digits of length up to 10 in its decimal expansion.
On ternary cubic forms.
On ternary inclusion-exclusion polynomials.
On Ternary Integral Recurrences
We prove that if a,b,c,d,e,m are integers, m > 0 and (m,ac) = 1, then there exist infinitely many positive integers n such that m|(an+b)cⁿ - deⁿ. Hence we derive a similar conclusion for ternary integral recurrences.
On ternary quadratic forms over the rational numbers
For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.
On the -class group of some number fields with large degree
Let be an odd square-free integer, any integer and . In this paper, we shall determine all the fields having an odd class number. Furthermore, using the cyclotomic -extensions of some number fields, we compute the rank of the -class group of whenever the prime divisors of are congruent to or .
On the -th power mean of with the weight of Gauss sums
The main purpose of this paper is to study the hybrid mean value of and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value of and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.
On the 2kth power mean of the character sums over short intervals
On the 2-primary part of a conjecture of Birch and Tate
On the 2-primary part of K₂ of rings of integers in certain quadratic number fields
1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of . For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form , where the primes are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of is zero for such fields. In the course of proving...
On the 2-Sylow subgroup of the Hilbert kernel of of number fields
On the 3-class field tower of some biquadratic fields
On the (3N+1)-conjecture
On the 4-norm of an automorphic form
We prove the optimal upper bound where runs over an orthonormal basis of Maass cusp forms of prime level and bounded spectral parameter.
On the 4-rank of ideal class groups of quadratic function fields
On the 4-rank of tame kernels of quadratic number fields
On the abc conjecture.
On the abc conjecture in algebraic number fields