KnowraDiophantine equationLinked fromLinked fromThe 41 pages that link to Diophantine equation, each with the reason it gives.All 41Broader topic 3Related 12Narrower topic 26IntegerRelated: It asks when integer values satisfy equations exactly.Number theoryBroader topic: Integer-solution questions form a central strand of number theory, from simple equations to unsolved problems.Algebraic numberRelated: Questions about algebraic roots often lead to arithmetic constraints on polynomial solutions.Pierre de FermatRelated: Many problems Fermat posed ask whether integer solutions to particular equations exist.BrahmaguptaRelated: Brahmagupta’s work on integer solutions belongs to a problem tradition also associated with Diophantus.Existential quantificationBroader topic: Asking whether it has a solution is an existential claim over integers.FibonacciNarrower topic: Problems of integer solutions underpin the mathematical investigations in his Book of Squares.p-adic numberNarrower topic: Checking solutions over p-adic fields can constrain the rational solutions of arithmetic equations.DiophantusNarrower topic: Modern terminology honors Diophantus for his attention to equations with constrained solutions.Hermann MinkowskiNarrower topic: Minkowski’s geometric methods turn questions about integer solutions into questions about lattice points.Polynomial equationRelated: Restricting solutions to integers turns polynomial solving into a number-theoretic problem.Additive number theoryNarrower topic: Representation questions become Diophantine equations when the allowed summands are specified.IdealRelated: Ideals help organize divisibility constraints in integer equations.Pell equationNarrower topic: The Pell equation is a classic case of an equation restricted to integer solutions.Pythagorean tripleNarrower topic: Finding triples is an integer-solution problem for a quadratic equation.Rational pointNarrower topic: Its rational solutions are rational points on the variety defined by the equation.Hilbert's tenth problemNarrower topic: These are precisely the equations whose integer solvability the problem asks an algorithm to decide.Egyptian fractionNarrower topic: Finding Egyptian fraction representations is a Diophantine problem with denominator constraints.Algebraic number fieldNarrower topic: Number fields provide tools for studying integer solutions and factoring equations.Fermat's theorem on sums of two squaresNarrower topic: The theorem answers an integer-solvability question for p=x²+y².Infinite descentNarrower topic: Descent most often rules out or classifies integer solutions to such equations.Algebraic equationBroader topic: Restricting solutions to integers can make solvability fundamentally different.Arithmetic geometryNarrower topic: The subject generalizes classical questions about integer solutions to geometric settings.Catalan's conjectureNarrower topic: Its proof advances methods for equations requiring integer solutions.Collatz conjectureRelated: Cycle and trajectory questions can be translated into arithmetic constraints on integers.Julia RobinsonNarrower topic: Hilbert’s tenth problem asks for a uniform method to test integer solutions.abc conjectureNarrower topic: The conjecture would control solutions to many equations through their prime factors.Erdős–Straus conjectureNarrower topic: The conjecture is an infinite family of equations requiring positive-integer solutions.Fermat's right triangle theoremNarrower topic: The triangle's side lengths and square area impose simultaneous integer equations.Ramanujan–Nagell equationNarrower topic: The equation belongs to the tradition of integer-solution problems named for Diophantus.Siegel's lemmaRelated: The lemma produces small integer relations that can constrain or construct solutions.Lifting-the-exponent lemmaRelated: Valuation bounds from LTE can rule out integer solutions to power equations.Markov numberNarrower topic: The defining equation makes Markov numbers part of an integer-solution problem.Sophie Germain's identityRelated: Integer substitutions turn the identity into constraints on solutions and divisibility.Sophie Germain's theoremNarrower topic: The theorem excludes integer solutions to a particular Diophantine equation.Beal's conjectureNarrower topic: Beal's conjecture asks what divisibility must hold for integer solutions to a power equation.Birch's theoremNarrower topic: The theorem counts integer solutions to a family of Diophantine equations.Bruck–Ryser–Chowla theoremRelated: The theorem’s obstructions depend on whether a parameter-defined equation has integer solutions.Goormaghtigh conjectureNarrower topic: The conjecture asks whether an integer equation has any solutions beyond two known cases.Squared triangular numberRelated: Finding these numbers amounts to solving for integers whose triangular number is a square.Tijdeman's theoremNarrower topic: Pairs of powers with a fixed difference form an integer equation.
KnowraDiophantine equationLinked fromLinked fromThe 41 pages that link to Diophantine equation, each with the reason it gives.All 41Broader topic 3Related 12Narrower topic 26IntegerRelated: It asks when integer values satisfy equations exactly.Number theoryBroader topic: Integer-solution questions form a central strand of number theory, from simple equations to unsolved problems.Algebraic numberRelated: Questions about algebraic roots often lead to arithmetic constraints on polynomial solutions.Pierre de FermatRelated: Many problems Fermat posed ask whether integer solutions to particular equations exist.BrahmaguptaRelated: Brahmagupta’s work on integer solutions belongs to a problem tradition also associated with Diophantus.Existential quantificationBroader topic: Asking whether it has a solution is an existential claim over integers.FibonacciNarrower topic: Problems of integer solutions underpin the mathematical investigations in his Book of Squares.p-adic numberNarrower topic: Checking solutions over p-adic fields can constrain the rational solutions of arithmetic equations.DiophantusNarrower topic: Modern terminology honors Diophantus for his attention to equations with constrained solutions.Hermann MinkowskiNarrower topic: Minkowski’s geometric methods turn questions about integer solutions into questions about lattice points.Polynomial equationRelated: Restricting solutions to integers turns polynomial solving into a number-theoretic problem.Additive number theoryNarrower topic: Representation questions become Diophantine equations when the allowed summands are specified.IdealRelated: Ideals help organize divisibility constraints in integer equations.Pell equationNarrower topic: The Pell equation is a classic case of an equation restricted to integer solutions.Pythagorean tripleNarrower topic: Finding triples is an integer-solution problem for a quadratic equation.Rational pointNarrower topic: Its rational solutions are rational points on the variety defined by the equation.Hilbert's tenth problemNarrower topic: These are precisely the equations whose integer solvability the problem asks an algorithm to decide.Egyptian fractionNarrower topic: Finding Egyptian fraction representations is a Diophantine problem with denominator constraints.Algebraic number fieldNarrower topic: Number fields provide tools for studying integer solutions and factoring equations.Fermat's theorem on sums of two squaresNarrower topic: The theorem answers an integer-solvability question for p=x²+y².Infinite descentNarrower topic: Descent most often rules out or classifies integer solutions to such equations.Algebraic equationBroader topic: Restricting solutions to integers can make solvability fundamentally different.Arithmetic geometryNarrower topic: The subject generalizes classical questions about integer solutions to geometric settings.Catalan's conjectureNarrower topic: Its proof advances methods for equations requiring integer solutions.Collatz conjectureRelated: Cycle and trajectory questions can be translated into arithmetic constraints on integers.Julia RobinsonNarrower topic: Hilbert’s tenth problem asks for a uniform method to test integer solutions.abc conjectureNarrower topic: The conjecture would control solutions to many equations through their prime factors.Erdős–Straus conjectureNarrower topic: The conjecture is an infinite family of equations requiring positive-integer solutions.Fermat's right triangle theoremNarrower topic: The triangle's side lengths and square area impose simultaneous integer equations.Ramanujan–Nagell equationNarrower topic: The equation belongs to the tradition of integer-solution problems named for Diophantus.Siegel's lemmaRelated: The lemma produces small integer relations that can constrain or construct solutions.Lifting-the-exponent lemmaRelated: Valuation bounds from LTE can rule out integer solutions to power equations.Markov numberNarrower topic: The defining equation makes Markov numbers part of an integer-solution problem.Sophie Germain's identityRelated: Integer substitutions turn the identity into constraints on solutions and divisibility.Sophie Germain's theoremNarrower topic: The theorem excludes integer solutions to a particular Diophantine equation.Beal's conjectureNarrower topic: Beal's conjecture asks what divisibility must hold for integer solutions to a power equation.Birch's theoremNarrower topic: The theorem counts integer solutions to a family of Diophantine equations.Bruck–Ryser–Chowla theoremRelated: The theorem’s obstructions depend on whether a parameter-defined equation has integer solutions.Goormaghtigh conjectureNarrower topic: The conjecture asks whether an integer equation has any solutions beyond two known cases.Squared triangular numberRelated: Finding these numbers amounts to solving for integers whose triangular number is a square.Tijdeman's theoremNarrower topic: Pairs of powers with a fixed difference form an integer equation.