On the representation of numbers as the sums.of the powers of primes
On the representation of the integer by positive quadratic forms with square-free variables
On the representations of a number as a sum of squares
On the residue mod 2 and mod 4 of p(n)
On the restricted Waring problem over
1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, , each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is a restricted...
On the seventh order mock theta functions.
On the sphere problem.
One of the oldest problems in analytic number theory consists of counting points with integer coordinates in the d-dimensional ball. It is very easy to find a main term for the counting function, but the size of the error term is difficult to estimate (...).
On the sum of a prime and the kth power of a prime
On the sum of two squares and two powers of k
It can be shown that the positive integers representable as the sum of two squares and one power of k (k any fixed integer ≥ 2) have positive density, from which it follows that those integers representable as the sum of two squares and (at most) two powers of k also have positive density. The purpose of this paper is to show that there is an infinity of positive integers not representable as the sum of two squares and two (or fewer) powers of k, k again any fixed integer ≥ 2.
On the sumset of the primes and a linear recurrence
Romanoff (1934) showed that integers that are the sum of a prime and a power of 2 have positive lower asymptotic density in the positive integers. We adapt his method by showing more generally the existence of a positive lower asymptotic density for integers that are the sum of a prime and a term of a given nonconstant nondegenerate integral linear recurrence with separable characteristic polynomial.
On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus
For and any sufficiently large odd we show that for almost all there exists a representation with primes mod for almost all admissible triplets of reduced residues mod .
On the transformation equation for the partition generating function.
On the upper bound for π₂(x)
On the function.
On the Vinogradov bound in the three primes Goldbach conjecture
On the Waring-Goldbach problem for one square and five cubes in short intervals
Let be a sufficiently large integer. We prove that almost all sufficiently large even integers can be represented as where with .
On the Waring-Goldbach problem with small non-integer exponent
On Vinogradov's estimate of trigonometric sums and the Goldbach-Vinogradov theorem
On Waring's problem
On Waring's problem (elementary methods).