Waring's problem

E451523

additive number theory problem number theory problem

Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.

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Observed surface forms (3)

Surface form	Occurrences
Waring's problem for k-th powers	1
Waring's theorem	1
Waring’s problem	1

Statements (53)

Predicate	Object
instanceOf	additive number theory problem ⓘ number theory problem ⓘ
asksFor	minimal number of k-th powers needed to represent any natural number ⓘ
countryOfOrigin	United Kingdom ⓘ
field	additive number theory ⓘ number theory ⓘ
focusesOn	bounds on number of summands ⓘ fixed exponents ⓘ
hasConjecture	exact values of G(k) for all k ⓘ exact values of g(k) for all k ⓘ
hasGeneralForm	for every k ≥ 2 there exists g(k) such that every natural number is a sum of at most g(k) k-th powers ⓘ
hasInvariant	G(k) ⓘ g(k) ⓘ
hasParameter	exponent k ⓘ
hasVariant	G(k) problem ⓘ Waring's problem for primes NERFINISHED ⓘ Waring–Goldbach problem NERFINISHED ⓘ asymptotic Waring's problem ⓘ g(k) problem ⓘ
influenced	development of additive number theory ⓘ methods in analytic number theory ⓘ
involvesConcept	existence of finite bounds ⓘ k-th powers of natural numbers ⓘ natural numbers ⓘ upper bounds ⓘ
involvesFunction	g(k) ⓘ
knownValueOfG(k)	G(2) = 4 ⓘ G(3) = 9 ⓘ G(4) = 16 ⓘ G(5) = 37 ⓘ
knownValueOfg(k)	g(2) = 4 ⓘ g(3) = 9 ⓘ g(4) = 19 ⓘ g(5) = 37 ⓘ
mainSubject	representation of natural numbers as sums of powers ⓘ
namedAfter	Edward Waring NERFINISHED ⓘ
originalFormulationLanguage	Latin ⓘ
proposer	Edward Waring NERFINISHED ⓘ
relatedTo	Hilbert's basis theorem (historical context) NERFINISHED ⓘ Hilbert's solution of Waring's problem ⓘ Lagrange's four-square theorem NERFINISHED ⓘ
solutionMethod	non-constructive existence proof ⓘ
solutionYear	1909 ⓘ
solvedBy	David Hilbert NERFINISHED ⓘ
specialCase	cubes ⓘ fourth powers ⓘ squares ⓘ
statedInWork	Meditationes Algebraicae NERFINISHED ⓘ
studiedBy	G. H. Hardy NERFINISHED ⓘ Klaus Roth NERFINISHED ⓘ R. C. Vaughan NERFINISHED ⓘ Srinivasa Ramanujan NERFINISHED ⓘ
yearProposed	1770 ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hardy–Littlewood circle method → appliedTo → Waring's problem ⓘ

Lagrange's four-square theorem → hasGeneralization → Waring's problem ⓘ

this entity surface form: Waring's problem for k-th powers

Fermat polygonal number theorem → relatedConcept → Waring's problem ⓘ

this entity surface form: Waring’s problem

Lagrange's four-square theorem → relatedTo → Waring's problem ⓘ

this entity surface form: Waring's theorem

Lagrange's four-square theorem → specialCaseOf → Waring's problem ⓘ