Abstract
The main purpose of this paper is to use the Hardy–Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.
Highlights
Introduction and Main ResultLet N, k1, k2, . . . , k s be natural numbers which satisfy 2 6 k1 6 k2 6 · · · 6 k s, N > s
K1 = k2 = · · · = k s, an interesting problem is to determine the value for k > 2, called Waring’s problem, of the function G (k), the least positive number s such that every sufficiently large number can be represented the sum of at most s k-th powers of natural numbers
The majority of information for G (k) has been derived from the Hardy–Littlewood method. This method has arised from a celebrated paper of Hardy and Ramanujan [3], which focused on the partition function
Summary
Let N, k1 , k2 , . . . , k s be natural numbers which satisfy 2 6 k1 6 k2 6 · · · 6 k s , N > s. 1 + k 2 + · · · + k s > 2, a standard application of the Hardy–Littlewood method suggests that all the integers, which satisfy necessary congruence conditions, could be written in the form (1). In view of the results of Brüdern and Kawada [20], Zhao [21], Liu and Lü [22] and Lü [23], it is reasonable to conjecture that, for sufficiently large integer N satisfying N ≡ 0 ( mod2), the following. Let E( N ) denote the number of positive integers n, which satisfy n ≡ 0 ( mod2), up to N, which can not be represented as n = p1 + p22 + p33 + p44. N is a sufficiently large integer and n ∈
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