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

Read more

Summary

Introduction and Main Result

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 ∈

Outline of the Proof of Theorem 1
Some Auxiliary Lemmas
The Singular Series
Proof of Proposition 2
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.