On the Representations of a Number as the Sum of Four Fifth Powers

It is known from Vaughan and Wooley's work on Waring's problemthat every sufficiently large natural number is the sum of atmost 17 fifth powers [13]. It is also known that at least sixfifth powers are required to be able to express every sufficientlylarge natural number as a sum of fifth powers (see, for instance,[5, Theorem 394]). The techniques of [13] allow one to showthat almost all natural numbers are the sum of nine fifth powers.A problem of related interest is to obtain an upper bound forthe number of representations of a number as a sum of a fixednumber of powers. Let R(n) denote the number of representationsof the natural number n as a sum of four fifth powers. In thispaper, we establish a non-trivial upper bound for R(n), whichis expressed in the following theorem.     

Slim Exceptional Sets For Sums of Four Squares

Given that available technology permits one to establish thatalmost all natural numbers satisfying appropriate congruenceconditions are represented as the sum of three squares of primenumbers, one expects strong estimates to be attainable for exceptionalsets in the analogous problem involving sums of four squaresof primes. Let E(N) denote the number of positive integers notexceeding N that are congruent to 4 modulo 24, yet cannot bewritten as the sum of four squares of prime numbers. A methodis described that shows that for each positive number , onehas , thereby exploiting effectively the ‘excess’ fourth square of a prime so as to improvethe recent bound due to J. Liu and M.-C. Liu. It transpires that the ideas underlying thisprogress permit estimates for exceptional sets in a varietyof additive problems to be significantly slimmed whenever sufficientlymany excess variables are available. Such ideas are illustratedfor several additional problems involving sums of four squares. 2000 Mathematical Subject Classification: 11P32, 11P05, 11P55.     

On a Waring-Goldbach-type problem for fourth powers

In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers.     

On Waring's Problem for Cubes and Smooth Weyl Sums

Non-trivial estimates for fractional moments of smooth cubicWeyl sums are developed. Complemented by bounds for such sumsof use on both the major and minor arcs in a Hardy-Littlewooddissection, these estimates are applied to derive an upper boundfor the sth moment of the smooth cubic Weyl sum of the expectedorder of magnitude as soon as s> 7.691. Related argumentsdemonstrate that all large integers n are represented as thesum of eight cubes of natural numbers, all of whose prime divisorsare at most exp (c(log nlog log n)^1/2}, for a suitable positivenumber c. This conclusion improves a previous result of G. Harcosin which nine cubes are required. 1991 Mathematics Subject Classification:11P05, 11L15, 11P55.     

On the Waring-Goldbach problem for seventh powers

We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of seventh powers of prime numbers.

     

Hua’s theorem with <Emphasis Type="Italic">s</Emphasis> almost equal prime variables

It is proved unconditionally that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented as the sum of s almost equal k-th powers of prime numbers for 2 ≤ k ≤ 10 and s =2k ＋ 1, which gives a short interval version of Hun＇s theorem.     

Waring-Goldbach Problem: One Square and Nine Biquadrates

In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers of prime numbers.     

Two results on powers of 2 in Waring-Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.     

On unequal powers of primes and powers of 2

We consider Linnik’s type of the Waring–Goldbach problem with unequal powers of primes. In particular, it is proved that every sufficiently large even integer can be represented as a sum of one prime, one square of prime, one cube of prime, one fourth power of prime and 18 powers of 2.     

On unlike powers of primes and powers of 2

It is proved that every sufficiently large even integer is a sum of one prime, one square of prime, two cubes of primes and 161 powers of 2.     

On the Exceptional Set of the Sum of a Prime Number and a Fixed Degree of a Prime Number

Let X be a sufficiently great real number and M denote the set of natural numbers not exceeding X which cannot be written as a sum of a prime and a fixed degree of a prime number from the arithmetical progression with difference d. Let Ed(X) = cardM. We obtain a new numerical degree estimate for the set Ed(X) and an estimate from below for the number of presentations of n ∉ M in the specified type. The proven estimates refine the generalization for an arithmetical progression of results earlier got by V.A. Plaksin.     

One Prime,Two Squares of Primes and Powers of 2

It is proved that every sufficiently large odd integer is the sum of one prime, two squares of primes and 35 powers of 2. This improves a previous result with 35 replaced by 83.     

On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors

The Ramanujan Journal - In this paper we prove that every sufficiently large odd integer can be written as a sum of a prime and 2 times a product of at most two distinct odd primes. Together with...     

A Three Squares Theorem with almost Primes

As an application of the vector sieve and uniform estimateson the Fourier coefficients of cusp forms of half-integral weight,it is shown that any sufficiently large number n 3 (mod 24)with 5 n is expressible as a sum of three squares of integershaving at most 521 prime factors. 2000 Mathematics Subject Classification11P05, 11N36, 11N75, 11E25.     

An Additive Problem with Piatetski-Shapiro Primes and Almost-Primes

Suppose that . We prove a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes p = [n ^1/ and show that every sufficiently large even integer can be written as a sum of a Piatetski-Shapiro prime and an almost-prime.Received November 29, 2001; in revised form August 21, 2002 Published online October 15, 2003     

The number of powers of 2 in a representation of large even integers (I)

Under the Generalized Riemann Hypothesis, it is proved that for any integer k⩾770 there is N_k>0 depending onk only such that every even integer ⩾N_k is a sum of two odd prime numbers andk powers of 2. The research is partially supported by RGC research grant (HKU 518/96P). The first author is supported by Post-Doctoral Fellowship of The University of Hong Kong.     

Multiplicative relations with conjugate algebraic numbers

We study what algebraic numbers can be represented by a product of algebraic numbers conjugate over a fixed number field K in fixed integer powers. The problem is nontrivial if the sum of these integer powers is equal to zero. The norm of such a number over K must be a root of unity. We show that there are infinitely many algebraic numbers whose norm over K is a root of unity and which cannot be represented by such a product. Conversely, every algebraic number can be expressed by every sufficiently long product in algebraic numbers conjugate over K. We also construct nonsymmetric algebraic numbers, i.e., algebraic numbers such that no elements of the corresponding Galois group acting on the full set of their conjugates form a Latin square. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 890–900, July, 2007.     

Goldbach Conjecture and the least prime number in an arithmetic progression

In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened form of the Chowla Hypothesis imply that every sufficiently large even integer may be written as the sum of two distinct primes.     

On the Exponential Sum with Square-Free Numbers

A new estimate for the exponential sum with square-free numbersis established. This result is applied to the problem of findingthe number of representations of a large integer as a sum ofthree square-free numbers. 2000 Mathematics Subject Classification11L07, 11N36, 11P99.     

华林问题中的一些新结果

设Ｇ（ｋ）为所有充分大的正整数均可表示为不超过ｓ个正整数ｋ次方幂之和的最小ｓ．本文对Ｇ（１２），Ｇ（１３），Ｇ（１９）给出了新的估计．