Ramanujan–Nagell equation
The Diophantine equation x² + 7 = 2ⁿ, whose positive integer solutions are (1, 3), (3, 4), (5, 5), (11, 7), and (181, 15).
Quadratic fields: Number systems formed by adjoining a square root of a nonsquare rational number to the rational numbers. The equation can be studied through algebraic integers in the quadratic field generated by √−7.
Srinivasa Ramanujan: An Indian mathematician whose work ranged across number theory, infinite series, partitions, and continued fractions. Ramanujan conjectured the complete list of positive integer solutions.
Pell equation: An equation of the form x² − Dy² = 1, with D a positive nonsquare integer. Both are classical integer problems with quadratic expressions, but their structures and solution patterns differ.
(1, 3) solution: The solution x = 1, n = 3, satisfying 1² + 7 = 2³. This is the smallest positive solution and verifies the equation directly.
Algebraic number theory: The study of number fields, their rings of integers, and arithmetic properties such as factorization and ideal classes. Its proof uses factorization and arithmetic properties of integers in a number field.
B. M. Wilson: A British mathematician who published an early proof concerning the equation now bearing Ramanujan's name. Wilson supplied an early proof that Ramanujan's proposed list was complete.
Perfect square: An integer equal to the square of another integer. Each solution requires 2ⁿ − 7 to be a perfect square.
(3, 4) solution: The solution x = 3, n = 4, satisfying 3² + 7 = 2⁴. It is the next solution, with an even value of x and a consecutive exponent.
Unique factorization: The property that elements factor uniquely into irreducible elements, up to units and order. Factoring expressions involving x and √−7 requires care because unique factorization can fail.
Trygve Nagell: A Norwegian mathematician known for research in number theory, especially Diophantine equations. Nagell gave a proof of the equation's complete solution set.