On waring-goldbach problems of small degrees

Abstract

For each fixed positive integer $k$, there exists a positive integer $s$ such that every sufficiently large positive integer $N$ satisfying certain necessary congruence conditions is the sum of $s$ $k$th powers of prime numbers. The classical problem of determining the smallest $s$ for which this is true is known as the Waring-Goldbach problem. In this Thesis, some generalizations and related results for the small degree cases $k=1,2$ and $3$ will be studied.

Let $a_1,a_2$ and $a_3$ be relatively prime positive integers. As a generalization of the weak Goldbach problem, it can be shown that the ternary linear equation $a_1p_1+a_2p_2+a_3p_3=b$ is solvable for all sufficiently large $b$ satisfying certain local conditions. It is an interesting question to obtain a lower bound of $b$ for which such a solution must exist. In this Thesis, an improved result of this lower bound will be obtained.

To approximate the Goldbach Conjecture, the representation of an even integer $N$ in the form $p_1+p_2+2^{\nu_1}+2^{\nu_2}+\cdots+2^{\nu_K}$ has been studied in the literature. As a modification, the new representation of $N$ in the form $p_1+p_2+b_12^{\nu_1}+b_22^{\nu_2}+\cdots+b_K2^{\nu_K}$ with bounded variables $b_1,b_2,\ldots,b_K$ will be studied in this Thesis. This provides an improvement to the approximation in terms of the density since a much smaller number of powers of $2$ is sufficient.

For degree $2$, it is widely conjectured that every sufficiently large positive integer $N=24k+4$ is a sum of four squares of prime numbers. This motivates the study of the Lagrange Equation $N=x_1^2+x_2^2+x_3^2+x_4^2$ with almost-prime variables $x_1,x_2,x_3$ and $x_4$. Results in various aspects will be obtained in this Thesis. Firstly, it will be proved that a solution exists with $x_1$ being a prime, $x_2$ being a prime or a product of two primes, and $x_3x_4$ having a bounded number of prime divisors. Secondly, when $x_1$ is a prime number, there exists a solution for which the total number of prime divisors of $x_2x_3x_4$ is at most $12$. Thirdly, with the use of a completely different sieve method, it can be shown that there exists a solution with $x_1,x_2$ and $x_3$ having at most $3$ prime divisors, and $x_4$ having at most $4$ prime divisors.

A mixed power equation involving cubes of primes and powers of $2$ will be studied. Part of the work involved can also be used to obtain the proportion of positive integers that can be represented as the sum of four cubes of primes. Both results obtained in this Thesis give considerable quantitative improvements over existing results.

In addition, the solvability of a kind of equation involving squares and cubes of integers belonging to subsets of primes with sufficient densities will be investigated. A Roth-type theorem will be obtained in this Thesis. The main ingredient of the proof is the use of the transference principle. The same technique can be applied to obtain analogous results for similar kind of mixed power equations of higher degrees.