Hall's conjecture

Last updated October 09, 2024

In mathematics, Hall's conjecture is an open question 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,

|y^{2}-x^{3}|>C{\sqrt {|x|}}.

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 the complex numbers C such that g(t)³ ≠ f(t)² in C[t], then

\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: 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,

|y^{2}-x^{3}|>c(\varepsilon )x^{1/2-\varepsilon }.

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.^[1] A generalization to other perfect powers is Pillai's conjecture, though it is also known that Pillai's conjecture would be true if Hall's conjecture held for any specific 0 < ε < 1/2.^[2]

The table below displays the known cases with $r={\sqrt {x}}/|y^{2}-x^{3}|>1$ . Note that y can be computed as the nearest integer to x^3/2. This list is known to contain all examples with $x\leq 10^{29}$ (the first 44 entries in the table) but may be incomplete past that point.

#	x	r
1	2	1.41
2	5234	4.26	^{[lower-alpha 1]}
3	8158	3.76	^{[lower-alpha 1]}
4	93844	1.03	^{[lower-alpha 1]}
5	367806	2.93	^{[lower-alpha 1]}
6	421351	1.05	^{[lower-alpha 1]}
7	720114	3.77	^{[lower-alpha 1]}
8	939787	3.16	^{[lower-alpha 1]}
9	28187351	4.87	^{[lower-alpha 1]}
10	110781386	1.23	^{[lower-alpha 1]}
11	154319269	1.08	^{[lower-alpha 1]}
12	384242766	1.34	^{[lower-alpha 1]}
13	390620082	1.33	^{[lower-alpha 1]}
14	3790689201	2.20	^{[lower-alpha 1]}
15	65589428378	2.19	^{[lower-alpha 2]}
16	952764389446	1.15	^{[lower-alpha 2]}
17	12438517260105	1.27	^{[lower-alpha 2]}
18	35495694227489	1.15	^{[lower-alpha 2]}
19	53197086958290	1.66	^{[lower-alpha 2]}
20	5853886516781223	46.60	^{[lower-alpha 2]}
21	12813608766102806	1.30	^{[lower-alpha 2]}
22	23415546067124892	1.46	^{[lower-alpha 2]}
23	38115991067861271	6.50	^{[lower-alpha 2]}
24	322001299796379844	1.04	^{[lower-alpha 2]}
25	471477085999389882	1.38	^{[lower-alpha 2]}
26	810574762403977064	4.66	^{[lower-alpha 2]}
27	9870884617163518770	1.90	^{[lower-alpha 3]}
28	42532374580189966073	3.47	^{[lower-alpha 3]}
29	44648329463517920535	1.79	^{[lower-alpha 3]}
30	51698891432429706382	1.75	^{[lower-alpha 3]}
31	231411667627225650649	3.71	^{[lower-alpha 3]}
32	601724682280310364065	1.88	^{[lower-alpha 3]}
33	4996798823245299750533	2.17	^{[lower-alpha 3]}
34	5592930378182848874404	1.38	^{[lower-alpha 3]}
35	14038790674256691230847	1.27	^{[lower-alpha 3]}
36	77148032713960680268604	10.18	^{[lower-alpha 4]}
37	180179004295105849668818	5.65	^{[lower-alpha 4]}
38	372193377967238474960883	1.33	^{[lower-alpha 3]}
39	664947779818324205678136	16.53	^{[lower-alpha 3]}
40	2028871373185892500636155	1.14	^{[lower-alpha 4]}
41	10747835083471081268825856	1.35	^{[lower-alpha 3]}
42	37223900078734215181946587	1.38	^{[lower-alpha 3]}
43	69586951610485633367491417	1.22	^{[lower-alpha 5]}
44	3690445383173227306376634720	1.51	^{[lower-alpha 3]}
45	133545763574262054617147641349	1.69	^{[lower-alpha 5]}
46	162921297743817207342396140787	10.65	^{[lower-alpha 5]}
47	374192690896219210878121645171	2.97	^{[lower-alpha 5]}
48	401844774500818781164623821177	1.29	^{[lower-alpha 5]}
49	500859224588646106403669009291	1.06	^{[lower-alpha 5]}
50	1114592308630995805123571151844	1.04	^{[lower-alpha 6]}
51	39739590925054773507790363346813	3.75	^{[lower-alpha 5]}
52	862611143810724763613366116643858	1.10	^{[lower-alpha 5]}
53	1062521751024771376590062279975859	1.006	^{[lower-alpha 5]}
54	6078673043126084065007902175846955	1.03	^{[lower-alpha 3]}

1 2 3 4 5 6 7 8 9 10 11 12 13 J. Gebel, A. Pethö and H.G. Zimmer.
1 2 3 4 5 6 7 8 9 10 11 12 Noam D. Elkies (including entry 16 which Elkies found but omitted from his published table).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 I. Jiménez Calvo, J. Herranz and G. Sáez (with the order of entries 29 and 30 corrected)
1 2 3 Johan Bosman (using the software of JHS).
1 2 3 4 5 6 7 8 9 S. Aanderaa, L. Kristiansen and H.K. Ruud.
↑ L.V. Danilov. Item 50 belongs to the infinite sequence found by Danilov.

Related Research Articles

<span class="mw-page-title-main">Diophantine equation</span> Polynomial equation whose integer solutions are sought

In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.

In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers $n$ and $k$ greater than 1, if the sum of $n$ many $k$ th powers of positive integers is itself a $k$ th power, then $n$ is greater than or equal to $k$ :

Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 2³ and 3² are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that

In mathematics, a Diophantine equation is an equation of the form P(x₁, ..., x_j, y₁, ..., y_k) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x₁, ..., x_j indicate parameters and y₁, ..., y_k indicate unknowns.

Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.

The abc conjecture is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers and that are relatively prime and satisfy . The conjecture essentially states that the product of the distinct prime factors of is usually not much smaller than . A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".

In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It is named for Trygve Nagell and Élisabeth Lutz.

In mathematics, a Thue equation is a Diophantine equation of the form

<span class="mw-page-title-main">Congruent number</span>

In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.

The Beal conjecture is the following conjecture in number theory:

In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.

A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation

In algebra, a Mordell curve is an elliptic curve of the form y² = x³ + n, where n is a fixed non-zero integer.

Brocard's problem is a problem in mathematics that seeks integer values of $such that is a perfect square, where is the factorial. Only three values of are known — 4, 5, 7 — and it is not known whether there are any more.$

In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem.

An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication. While in number theory they have important consequences in the solving of Diophantine equations, with respect to cryptography, they enable us to make effective use of the difficulty of the discrete logarithm problem (DLP) for the group $, of elliptic curves over a finite field, where q = p k and p is a prime. The DLP, as it has come to be known, is a widely used approach to public key cryptography, and the difficulty in solving this problem determines the level of security of the cryptosystem. This article covers algorithms to count points on elliptic curves over fields of large characteristic, in particular p > 3. For curves over fields of small characteristic more efficient algorithms based on p -adic methods exist.$

In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form

In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve $defined over the field of rational numbers or more generally a number field K . Mordell's theorem says the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is the rank of the curve.$

In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for an integer $to equal such a sum is that cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and -1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient.$

References

↑ Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. pp. 205–206. ISBN 3-540-54058-X. Zbl 0754.11020.
↑ Nair, M (1 December 1977). "A NOTE ON THE EQUATION x^3−y^2=k". The Quarterly Journal of Mathematics. 29 (4): 483–487.

Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. D9. ISBN 978-0-387-20860-2. Zbl 1058.11001.
Hall, Jr., Marshall (1971). "The Diophantine equation x³ - y² = k". In Atkin, A.O.L.; Birch, B. J. (eds.). Computers in Number Theory. pp. 173–198. ISBN 0-12-065750-3. Zbl 0225.10012.
Elkies, N.D. "Rational points near curves and small nonzero | 'x³ - y²'| via lattice reduction", http://arxiv.org/abs/math/0005139
Danilov, L.V., "The Diophantine equation 'x³ - y²' ' = k ' and Hall's conjecture", 'Math. Notes Acad. Sci. USSR' 32(1982), 617-618.
Gebel, J., Pethö, A., and Zimmer, H.G.: "On Mordell's equation", 'Compositio Math.' 110(1998), 335-367.
I. Jiménez Calvo, J. Herranz and G. Sáez Moreno, "A new algorithm to search for small nonzero |'x3 - y2'| values", 'Math. Comp.' 78 (2009), pp. 2435-2444.
S. Aanderaa, L. Kristiansen and H. K. Ruud, "Search for good examples of Hall's conjecture", 'Math. Comp.' 87 (2018), 2903-2914.

External links

a page on the problem by Noam Elkies

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[GPZ-3] 1 2 3 4 5 6 7 8 9 10 11 12 13 J. Gebel, A. Pethö and H.G. Zimmer.

[E-4] 1 2 3 4 5 6 7 8 9 10 11 12 Noam D. Elkies (including entry 16 which Elkies found but omitted from his published table).

[JHS-5] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 I. Jiménez Calvo, J. Herranz and G. Sáez (with the order of entries 29 and 30 corrected)

[JB-6] 1 2 3 Johan Bosman (using the software of JHS).

[AKR-7] 1 2 3 4 5 6 7 8 9 S. Aanderaa, L. Kristiansen and H.K. Ruud.

[D-8] L.V. Danilov. Item 50 belongs to the infinite sequence found by Danilov.

[1] Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. pp. 205–206. ISBN 3-540-54058-X. Zbl 0754.11020.

[2] Nair, M (1 December 1977). "A NOTE ON THE EQUATION x^3−y^2=k". The Quarterly Journal of Mathematics. 29 (4): 483–487.

[1]

[2]

[lower-alpha 1]

[lower-alpha 2]

[lower-alpha 3]

[lower-alpha 4]

[lower-alpha 5]

[lower-alpha 6]