Diophantine equations

1 article

Linear diophantine equations Pell equation Fermats last theorem Archimedess cattle problem Beal conjecture Bhaskaras lemma Brahmaguptas problem Brauers theorem on forms Brown numbers Cannonball problem Catalans conjecture Chakravala method Chevalley warning theorem Coin problem Congruum Cuboid conjectures Diagonal form Diophantine quintuple Eisenstein triple Equation xy yx Erdos diophantine graph Erdos graham conjecture Erdos moser equation Erdos straus conjecture Eulers sum of powers conjecture Fermat catalan conjecture Fermats right triangle theorem Formulas for generating pythagorean triples Goormaghtigh conjecture Heegners lemma Hilberts tenth problem Infinite descent Jacobi madden equation Lander parkin and selfridge conjecture Legendres equation Lonely runner conjecture Markov number Norm form Optic equation Pillais conjecture Polynomial diophantine equation Prouhet tarry escott problem Pythagorean triple Ramanujan nagell equation Runges method S unit equation Siegel identity Sum of four cubes problem Thue equation Tijdemans theorem Tree of primitive pythagorean triples Tunnells theorem Vieta jumping Znams problem Abs methods Alpha max plus beta min algorithm Andr oort conjecture Approximation in algebraic groups Arakelov theory Arithmetic of abelian varieties Arithmetic surface B zouts identity Bogomolov conjecture Bombieri lang conjecture Boolean pythagorean triples problem Brides chair British flag theorem Conductor of an abelian variety Diophantine geometry Diophantus and diophantine equations Diophantus ii viii Erd's anning theorem Erd's diophantine graph Erd's moser equation Erd's straus conjecture Erd's ulam problem Euclidean distance Euler brick Faltings theorem Fermat curve Field of definition Garfields proof of the pythagorean theorem Harborths conjecture Hasse principle Height zeta function Heronian tetrahedron Heronian triangle Hundred fowls problem Hypotenuse Integer lattice Integer triangle Kaprekar number Manin conjecture Manin obstruction Mordellic variety Nevanlinna invariant Nonhypotenuse number Perfect digital invariant Principal homogeneous space Proof by infinite descent Pythagorean addition Pythagorean theorem Pythagorean triangle Pythagorean trigonometric identity Ramanujans ternary quadratic form Rational point Robbins pentagon Siegels lemma Siegels theorem on integral points Spiral of theodorus Square root of 2 Tate conjecture The monkey and the coconuts Thin set Torsion conjecture Xuan tu Zilber pink conjecture

Geometry LycéeExplication10 min

A Circle in a Square, a 5 by 10 Rectangle, and the Root You Throw Away

One radius drawn to the rectangle's far corner turns the whole problem into a right triangle with legs R - 10 and R - 5, and the quadratic that follows has roots 5 and 25. Both satisfy the equation exactly, so rejecting 5 takes geometry rather than arithmetic: at that radius the corner really does touch the circle while the rectangle has already swallowed half the disk. The general a by b rectangle shows the discarded root is a permanent feature of squaring.