Catalan’s conjecture

Target entity: Catalan’s conjecture
Context triple: [Beal conjecture, relatedTo, Catalan’s conjecture]

• A. Pillai’s conjecture
  Pillai’s conjecture is an unproven statement in number theory asserting that the difference between perfect powers takes each positive integer value only finitely many times.
• B. Erdős–Moser equation
  The Erdős–Moser equation is a famous unsolved Diophantine equation in number theory that asks whether 1^k + 2^k + ... + (m−1)^k = m^k has any integer solutions beyond the trivial case (k, m) = (1, 2).
• C. Fermat's theorem on sums of two squares
  Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
• D. Fermat polygonal number theorem
  The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
• E. 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.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Catalan’s conjecture
Target entity description: Catalan’s conjecture is a number theory statement, now proven as Mihăilescu’s theorem, asserting that 8 and 9 are the only consecutive perfect powers among the positive integers.

• A. Pillai’s conjecture
  Pillai’s conjecture is an unproven statement in number theory asserting that the difference between perfect powers takes each positive integer value only finitely many times.
• B. Erdős–Moser equation
  The Erdős–Moser equation is a famous unsolved Diophantine equation in number theory that asks whether 1^k + 2^k + ... + (m−1)^k = m^k has any integer solutions beyond the trivial case (k, m) = (1, 2).
• C. Fermat's theorem on sums of two squares
  Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
• D. Fermat polygonal number theorem
  The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
• E. 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.
• F. None of above. chosen