In mathematics, Hall's conjecture is an open question on the differences between perfect squares and perfect cubes. It asserts that a perfect square y 2 {\displaystyle y^{2}} and a perfect cube x 3 {\displaystyle x^{3}} that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves. The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant C {\displaystyle C} such that for any integers x {\displaystyle x} and y {\displaystyle y} for which y 2 ≠ x 3 {\displaystyle y^{2}\neq x^{3}} ,
| y 2 − x 3 | > C | x | 1 / 2 {\displaystyle |y^{2}-x^{3}|>C|x|^{1/2}} . Hall suggested that perhaps C {\displaystyle C} could be taken as 1 / 5 {\displaystyle 1/5} , which was consistent with all the data known at the time the conjecture was proposed. In 1982, Danilov proved that | y 2 − x 3 | {\displaystyle |y^{2}-x^{3}|} is less than 433 2 × | x | 1 / 2 {\displaystyle 433{\sqrt {2}}\times |x|^{1/2}} infinitely often, thus providing an upper bound on C {\displaystyle C} and proving that the exponent 1 / 2 {\displaystyle 1/2} (that is, the use of | x | 1 / 2 {\displaystyle |x|^{1/2}} ) is optimal and cannot be replaced by any higher power: for no δ > 0 {\displaystyle \delta >0} is there a constant C {\displaystyle C} such that | y 2 − x 3 | > C | x | 1 / 2 + δ {\displaystyle |y^{2}-x^{3}|>C|x|^{1/2+\delta }} whenever y 2 ≠ x 3 {\displaystyle y^{2}\neq x^{3}} . In 1965, Davenport proved an analogue of the above conjecture in the case of polynomials: if f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} are nonzero polynomials over the complex numbers C {\displaystyle \mathbb {C} } such that g ( t ) 3 ≠ f ( t ) 2 {\displaystyle g(t)^{3}\neq f(t)^{2}} in C [ t ] {\displaystyle \mathbb {C} [t]} , then
deg ( g ( t ) 2 − f ( t ) 3 ) ≥ 1 2 deg f ( t ) + 1. {\displaystyle \deg(g(t)^{2}-f(t)^{3})\geq {\frac {1}{2}}\deg f(t)+1.}
The weak form of Hall's conjecture, stated by Stark and Trotter around 1980, replaces the square root on the right side of the inequality by any exponent less than 1 / 2 {\displaystyle 1/2} : for any ε > 0 {\displaystyle \varepsilon >0} , there is some constant c ( ε ) {\displaystyle c(\varepsilon )} depending on ε {\displaystyle \varepsilon } such that for any integers x {\displaystyle x} and y {\displaystyle y} for which y 2 ≠ x 3 {\displaystyle y^{2}\neq x^{3}} ,
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