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- In mathematics, Hall's conjecture is an open question, as of 2015, on the differences between perfect squares and perfect cubes. It asserts that a perfect square y2 and a perfect cube x3 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 such that for any integers x and y for which y2 ≠ x3, Hall suggested that perhaps C could be taken as 1/5, which was consistent with all the data known at the time the conjecture was proposed. Danilov showed in 1982 that the exponent 1/2 on the right side (that is, the use of |x|1/2) cannot be replaced by any higher power: for no δ > 0 is there a constant C such that |y2 - x3| > C|x|1/2 + δ whenever y2 ≠ x3. In 1965, Davenport proved an analogue of the above conjecture in the case of polynomials: if f(t) and g(t) are nonzero polynomials over C such that g(t)3 ≠ f(t)2 in C[t], then 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: for any ε > 0, there is some constant c(ε) depending on ε such that for any integers x and y for which y2 ≠ x3, The original, strong, form of the conjecture with exponent 1/2 has never been disproved, although it is no longer believed to be true and the term Hall's conjecture now generally means the version with the ε in it. For example, in 1998, Noam Elkies found the example 4478849284284020423079182 - 58538865167812233 = -1641843, for which compatibility with Hall's conjecture would require C to be less than .0214 ≈ 1/50, so roughly 10 times smaller than the original choice of 1/5 that Hall suggested. The weak form of Hall's conjecture would follow from the ABC conjecture. A generalization to other perfect powers is Pillai's conjecture. The table below displays the known cases with . Note that y can be computed as thenearest integer to x3/2. (en)
- En matemáticas, la conjetura de Hall es una pregunta abierta, a partir de 2015, sobre las diferencias entre cuadrados perfectos y cubos perfectos. Afirma que un cuadrado perfecto y2 y un cubo perfecto x3 que no son iguales deben estar a una distancia sustancial entre sí. Esta pregunta surgió al considerar la ecuación de Mordell en la teoría de puntos enteros en curvas elípticas. (es)
- In matematica, la congettura di Marshall Hall è un problema aperto di teoria dei numeri sulla differenza tra quadrati perfetti e . Essa afferma che se e non sono uguali, allora la loro distanza deve essere superiore a una costante dipendente da . Questa congettura, che prende il nome dal matematico , deriva da alcune considerazioni sui punti interi della , nella teoria delle curve ellittiche. (it)
- Гипотеза Холла — нерешённая на 2015 г. теоретико-числовая гипотеза об оценке сверху для решений диофантова при заданном . Имеет несколько формулировок разной силы. Была сформулирована Холлом в 1971 г. (ru)
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- En matemáticas, la conjetura de Hall es una pregunta abierta, a partir de 2015, sobre las diferencias entre cuadrados perfectos y cubos perfectos. Afirma que un cuadrado perfecto y2 y un cubo perfecto x3 que no son iguales deben estar a una distancia sustancial entre sí. Esta pregunta surgió al considerar la ecuación de Mordell en la teoría de puntos enteros en curvas elípticas. (es)
- In matematica, la congettura di Marshall Hall è un problema aperto di teoria dei numeri sulla differenza tra quadrati perfetti e . Essa afferma che se e non sono uguali, allora la loro distanza deve essere superiore a una costante dipendente da . Questa congettura, che prende il nome dal matematico , deriva da alcune considerazioni sui punti interi della , nella teoria delle curve ellittiche. (it)
- Гипотеза Холла — нерешённая на 2015 г. теоретико-числовая гипотеза об оценке сверху для решений диофантова при заданном . Имеет несколько формулировок разной силы. Была сформулирована Холлом в 1971 г. (ru)
- In mathematics, Hall's conjecture is an open question, as of 2015, on the differences between perfect squares and perfect cubes. It asserts that a perfect square y2 and a perfect cube x3 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 such that for any integers x and y for which y2 ≠ x3, 4478849284284020423079182 - 58538865167812233 = -1641843, (en)
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- Conjetura de Hall (es)
- Hall's conjecture (en)
- Congettura di Marshall Hall (it)
- Гипотеза Холла (ru)
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