Marshall Hall's conjecture
Encyclopedia
In mathematics
, Marshall Hall's conjecture is an open question, , on the differences between perfect squares
and perfect cubes. Aside from the case of a perfect sixth power, it asserts that a perfect square m2 and a perfect cube n3 must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curve
s.
The weak form of Hall's conjecture is formulated as
where C(n) is an exponential factor less than 1, but which tends to 1 as n → ∞. That is, for any given ε > 0, we can assert that
The strong form, on which doubt has been cast, replaces the LHS with a constant multiple of
. This was the original formulation of Marshall Hall, Jr. in 1970.
The weak form of the conjecture would follow from the ABC conjecture
.
A generalization to other perfect powers is Pillai's conjecture.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Marshall Hall's conjecture is an open question, , on the differences between perfect squares
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
and perfect cubes. Aside from the case of a perfect sixth power, it asserts that a perfect square m2 and a perfect cube n3 must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s.
The weak form of Hall's conjecture is formulated as
where C(n) is an exponential factor less than 1, but which tends to 1 as n → ∞. That is, for any given ε > 0, we can assert that
The strong form, on which doubt has been cast, replaces the LHS with a constant multiple of
. This was the original formulation of Marshall Hall, Jr. in 1970.The weak form of the conjecture would follow from the ABC conjecture
Abc conjecture
The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c , which have no common factor and satisfy a + b = c...
.
A generalization to other perfect powers is Pillai's conjecture.
External links
- http://www.math.harvard.edu/~elkies/hall.html, page of Noam ElkiesNoam ElkiesNoam David Elkies is an American mathematician and chess master.At age 14, Elkies received a gold medal with a perfect score at the International Mathematical Olympiad, the youngest ever to do so...
on the problem. - http://ijcalvo.galeon.com/hall.htm, table of good examples of Marshall Hall's conjecture by Ismael Jimenez Calvo.