On a decomposition of polynomials in several variables
One considers representation of a polynomial in several variables as the sum of values of univariate polynomials taken at linear combinations of the variables.
On a decomposition of polynomials in several variables, II
One considers representation of cubic polynomials in several variables as the sum of values of univariate polynomials taken at linear combinations of the variables.
On an Asymptotic Formula of Ramanujan.
On another sieve method and the numbers that are a sum of two hth powers: II.
On binary quartic forms.
On certain additive representations of integers
On so that .
On sot hat represents and and not .
On Differences and sums of Integers, II
On pairs of binary forms with given resultant or given semi-resultant.
On powerful numbers.
On some arithmetical properties of weighted sums of S-units.
On some universal sums of generalized polygonal numbers
For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in...
On sums and products in a field
We study sums and products in a field. Let be a field with , where is the characteristic of . For any integer , we show that any can be written as with and , and that for any we can write every as with and . We also prove that for any and there are such that .
On sums of powers and a related problem
On the congruence
On the Diophantine equation
On the diophantine equations
On the Frobenius problem for geometric sequences.
On the numbers that are representable as the sum of two cubes.