Artificial intelligence

Mathematics

Computer science

Algorithm

Transactions of the American Mathematical Society

We develop a variety of new techniques to treat Diophantine equations of the shape <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x squared plus upper D equals y Superscript n"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">x^2+D =y^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, based upon bounds for linear forms in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantin...

Differences between perfect powers: The Lebesgue-Nagell equation