Pell-type equations

Target entity: Pell-type equations
Context triple: [Indian mathematics, hasNotableConcept, Pell-type equations]

• A. Lebesgue–Nagell equation
  The Lebesgue–Nagell equation is a Diophantine equation of the form \(x^2 + D = y^n\) (with fixed integers \(D\) and \(n \ge 3\)) studied in number theory for its finite and often explicitly determinable set of integer solutions.
• B. Diophantine equations
  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.
• C. Ramanujan–Nagell equation
  The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
• D. On Pythagorean Numbers
  On Pythagorean Numbers is a lost philosophical work by the ancient Greek philosopher Speusippus that explored numerical doctrines associated with Pythagorean thought.
• E. Baker theorem on linear forms in logarithms
  The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Pell-type equations
Target entity description: Pell-type equations are a class of quadratic Diophantine equations, typically of the form x² − Ny² = 1, that have been studied extensively in number theory since ancient times, including in Indian mathematics.

• A. Lebesgue–Nagell equation
  The Lebesgue–Nagell equation is a Diophantine equation of the form \(x^2 + D = y^n\) (with fixed integers \(D\) and \(n \ge 3\)) studied in number theory for its finite and often explicitly determinable set of integer solutions.
• B. Diophantine equations
  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.
• C. Ramanujan–Nagell equation
  The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
• D. chakravala method for solving indeterminate equations
  The chakravala method for solving indeterminate equations is an ancient Indian cyclic algorithm, notably used to solve Pell-type quadratic Diophantine equations with remarkable efficiency and generality.
• E. On Pythagorean Numbers
  On Pythagorean Numbers is a lost philosophical work by the ancient Greek philosopher Speusippus that explored numerical doctrines associated with Pythagorean thought.
• F. None of above. chosen