Diophantine equations

E629500

mathematical concept topic in number theory type of equation

Diophantine equations are polynomial equations for which only integer or rational solutions are sought, forming a central and often notoriously difficult area of number theory.

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Observed surface forms (2)

Surface form	Occurrences
Diophantine Equations	1
Thue-type equation	1

Statements (54)

Predicate	Object
instanceOf	mathematical concept ⓘ topic in number theory ⓘ type of equation ⓘ
centralIn	number theory ⓘ
decidabilityResult	no algorithm exists to decide solvability over integers (Hilbert's tenth problem) ⓘ
distinguishedFrom	complex-valued polynomial equations ⓘ real-valued polynomial equations ⓘ
field	number theory ⓘ
hasExample	Fermat equation x^n + y^n = z^n NERFINISHED ⓘ Markov equation x^2 + y^2 + z^2 = 3xyz NERFINISHED ⓘ Mordell equation y^2 = x^3 + k ⓘ Pell equation NERFINISHED ⓘ Pythagorean triple equation x^2 + y^2 = z^2 ⓘ Thue equation NERFINISHED ⓘ unit equation x + y = 1 in S-units ⓘ
hasSubcategory	Diophantine approximation ⓘ Diophantine inequalities ⓘ Diophantine sets NERFINISHED ⓘ exponential Diophantine equations ⓘ higher-degree Diophantine equations ⓘ linear Diophantine equations ⓘ quadratic Diophantine equations ⓘ
involves	polynomial equations ⓘ
knownFor	difficulty ⓘ
namedAfter	Diophantus of Alexandria NERFINISHED ⓘ
relatedTo	Birch and Swinnerton-Dyer conjecture NERFINISHED ⓘ Diophantine sets NERFINISHED ⓘ Faltings' theorem NERFINISHED ⓘ Fermat's Last Theorem NERFINISHED ⓘ Galois representations NERFINISHED ⓘ Hasse principle ⓘ Hilbert's tenth problem NERFINISHED ⓘ Matiyasevich's theorem NERFINISHED ⓘ Mordell conjecture NERFINISHED ⓘ Siegel's theorem NERFINISHED ⓘ abc conjecture NERFINISHED ⓘ elliptic curves ⓘ height functions ⓘ local-global principle ⓘ modular forms ⓘ rational points on varieties ⓘ
solutionDomain	integers ⓘ rational numbers ⓘ
solvabilityProperty	undecidable in general ⓘ
studiedIn	ancient Greek mathematics ⓘ modern mathematics ⓘ
subfieldOf	algebraic number theory ⓘ arithmetic geometry ⓘ
usesMethod	algebraic geometry ⓘ computational number theory ⓘ congruences ⓘ geometry of numbers ⓘ p-adic methods ⓘ transcendence theory ⓘ

Referenced by (6)

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

Ramanujan–Nagell equation → classification → Diophantine equations ⓘ

this entity surface form: Thue-type equation

Hilbert’s tenth problem → concerns → Diophantine equations ⓘ

Unsolved Problems in Number Theory → coversTopic → Diophantine equations ⓘ

Diophantine geometry → fieldOfStudy → Diophantine equations ⓘ

Louis Mordell → notableWork → Diophantine equations ⓘ

this entity surface form: Diophantine Equations

Fermat Prize → relatedTo → Diophantine equations ⓘ