Waring's Problem Research Articles

On Waring's problem: two cubes and seven biquadrates

On Waring's problem: two cubes and seven biquadrates

Sums of three cubes

MathematikaVolume 47, Issue 1-2 p. 53-61 Research Article Sums of three cubes Professor Trevor D. Wooley, Professor Trevor D. Wooley wooley@math.lsa.umich.edu Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI, 48109-1109 U.S.A.Search for more papers by this author Professor Trevor D. Wooley, Professor Trevor D. Wooley wooley@math.lsa.umich.edu Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI, 48109-1109 U.S.A.Search for more papers by this author First published: 01 December 2000 https://doi.org/10.1112/S0025579300015710Citations: 10AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume47, Issue1-2December 2000Pages 53-61 RelatedInformation

On a refinement of Waring's problem

On a refinement of Waring's problem

Waring's Problem with Polynomial Summands

Let f ( x ) be an integer-valued polynomial with no fixed integer divisor [ges ] 2, that is, for no integer d [ges ] 2 does d [mid ] f ( x ) for all integers x . One generalization of the famous Waring problem is to determine whether, for large enough s , the equation formula here is solvable in positive integers x 1 , …, x s for sufficiently large integers n . The existence of such s for every f was established by Kamke [ 5 ] in 1921. Subsequent authors (Pillai, Hua [ 2–4 ], Vinogradov, Nacaev [ 7 ], and others) have studied the problem of bounding G ( f ), the least s for which (1.1) is solvable for all large n . Questions of local solubility of (1.1), that is, solubility of the congruence formula here play a more important and complicated role in this problem than in the classical Waring problem. Let Γ 0 ( f ) denote the least number s so that (1.2) is solvable for every pair n , q . It is well known that Γ 0 ( x k ) [les ] 4 k for every k , but Hua [ 4 ] found that for every k , the polynomial formula here satisfies Γ 0 ( f k ) [ges ] 2 k − 1 (take s = 2 k −2, q = 2 k and n = (−1) k in (1.2)). Clearly G ( f ) [ges ] Γ 0 ( f ), but one can say more by restricting the values of n under consideration, as has been done by several authors in the case f ( x ) = x 4 (for example [ 1 , 6 ]). The singular series formula here where e ( z ) = e 2π iz , encapsulates the local solubility information. In particular, [Sfr ] s, f ( n ) [ges ] 0 for every n and [Sfr ] s, f ( n ) > 0 if and only if (1.2) is soluble for every q . Define G ( f ) to be the least number s so that for every δ >0 and every n > n 0 (δ) with [Sfr ] s, f ( n ) [ges ] δ, (1.1) is soluble. The reason for taking [Sfr ] s, f ( n ) [ges ] δ instead of [Sfr ] s, f ( n ) > 0 is that we wish to exclude from consideration certain n lying in sparse sequences for which (1.1) is insoluble but [Sfr ] s, f ( n ) > 0. For example, taking f ( x ) = x 4 , s = 15 and n j = 79·16 j ( j = 0, 1, …), it can be shown that (1.1) is not soluble for n = n j , that [Sfr ] s, f ( n j ) > 0 for all j , and that [Sfr ] s, f ( n j ) → 0 as j → ∞. It is known that G ( x 4 ) = 16 (see [ 1 ]) and that G ( x 4 ) [ges ] 11 almost holds (see [ 6 ]).

Further improvements in Waring's problem, IV: Higher powers

Further improvements in Waring's problem, IV: Higher powers

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

On Waring's problem for a prime modulus

We obtain a lower bound for the minimum over positive integers such that the sum of certain powers of some integers is divisible by a prime number, but none of these integers is divisible by this prime number.

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 problem that every sufficiently large natural number is the sum of at most 17 fifth powers [13]. It is also known that at least six fifth powers are required to be able to express every sufficiently large natural number as a sum of fifth powers (see, for instance, [5, Theorem 394]). The techniques of [13] allow one to show that almost all natural numbers are the sum of nine fifth powers. A problem of related interest is to obtain an upper bound for the number of representations of a number as a sum of a fixed number of powers. Let R(n) denote the number of representations of the natural number n as a sum of four fifth powers. In this paper, we establish a non-trivial upper bound for R(n), which is expressed in the following theorem.

On an identity of Hilbert

A simple proof of the polynomial identity used by Hilbert in the solution of the Waring problem is given. The proof is based on the continued fraction expansion of a certain formal hypergeometric series.

On Waring's problem: a square, four cubes and a biquadrate

Additive representations of natural numbers by mixtures of squares, cubes and biquadrates belong to the class of more interesting special cases which form the object of attention for testing the general expectation that any sufficiently large natural number n is representable in the formformula hereas soon as the reciprocal sum [sum ]sj=1k−1j is reasonably large. With the exception of a handful of very special problems, in the current state of knowledge the latter reciprocal sum must exceed 2, at the very least, in order that it be feasible to successfully apply the Hardy–Littlewood method to treat the corresponding additive problem. Here we remove a case from the list of those combinations of exponents which have defied treatment thus far.

On Waring's Problem in Number Fields

On Waring's Problem in Number Fields

On the p-adic Waring's problem

On the p-adic Waring's problem

On Waring's problem in finite fields

On Waring's problem in finite fields

On Waring's problem with polynomial summands II

On Waring's problem with polynomial summands II

On Waring's problem with quartic polynomial summands

On Waring's problem with quartic polynomial summands

On the sum of four cubes

On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

Some new results in Waring's problem

LetG(k) denote the smallest numbers such that every sufficiently large natural number is the sum of, at most,s k-th powers of natural numbers. In this paper, we give new bounds forG(12),G(13) andG(19).

The Hardy-Littlewood conjecture. An algebraic approach

We consider the well-known Hardy-Littlewood equation, which is related to the Waring problem, in terms of roots of polynomials of the corresponding degree. The Hardy-Littlewood conjecture is confirmed for cubic fields. Bibliography: 3 titles.

On Waring's problem with polynomial summands

On Waring's problem with polynomial summands

Limitation to the asymptotic formula in Waring's problem

Limitation to the asymptotic formula in Waring's problem