ABSTRACT
A straightforward consequence of Thue’s pioneering work on Diophantine approximation [26] is: Let m be a non-zero integer and let f ∈ Z [x,y] be a binary form. Then the equation https://www.w3.org/1998/Math/MathML"> f ( x , y ) = m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq620.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> has finitely many integer solutions, unless f is the multiple of either a power of a linear form, or a power of a binary quadratic form with positive nonsquare discriminant. Thue’s techniques were refined by Pólya [17] and Siegel [20] to show that if f ∈ Z[x] has at least two distinct roots, then () https://www.w3.org/1998/Math/MathML"> P ( f ( x ) ) → ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq621.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
