Hall’s conjecture – Wikipedia

In mathematics, Hall’s conjecture is an open question, as of 2015^[update], 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,

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

${displaystyle deg(g(t)^{2}-f(t)^{3})geq {frac {1}{2}}deg f(t)+1.}$

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: 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,

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

[1] A generalization to other perfect powers is Pillai’s conjecture.

The table below displays the known cases with

${displaystyle r={sqrt {x}}/|y^{2}-x^{3}|>1}$

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

1. ^ ^{a b c d e f g h i j k l m} J. Gebel, A. Pethö and H.G. Zimmer.
2. ^ ^{a b c d e f g h i j k l} Noam D. Elkies.
3. ^ ^{a b c d e f g h i j k l m n o} I. Jiménez Calvo, J. Herranz and G. Sáez.
4. ^ ^{a b c} Johan Bosman (using the software of JHS).
5. ^ ^{a b c d e f g h i} S. Aanderaa, L. Kristiansen and H.K. Ruud.
6. ^ L.V. Danilov. Item 50 belongs to the infinite sequence found by Danilov.

References[edit]

• 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[edit]