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,
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
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.
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. ISBN0-12-065750-3. Zbl 0225.10012.
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.
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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 y 2 x 3 gt C x displaystyle y 2 x 3 gt 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 d gt 0 is there a constant C such that y2 x3 gt C x 1 2 d 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 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 for any e gt 0 there is some constant c e depending on e such that for any integers x and y for which y2 x3 y 2 x 3 gt c e x 1 2 e displaystyle y 2 x 3 gt 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 e in it For example in 1998 Noam Elkies found the example4478849284284020423079182 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 The table below displays the known cases with r x y 2 x 3 gt 1 displaystyle r sqrt x y 2 x 3 gt 1 Note that y can be computed as the nearest integer to x3 2 x r1 2 1 412 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 a b c d e f g h i j k l m J Gebel A Petho and H G Zimmer a b c d e f g h i j k l Noam D Elkies a b c d e f g h i j k l m n o I Jimenez Calvo J Herranz and G Saez a b c Johan Bosman using the software of JHS a b c d e f g h i S Aanderaa L Kristiansen and H K Ruud L V Danilov Item 50 belongs to the infinite sequence found by Danilov References Edit 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 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 Petho A and Zimmer H G On Mordell s equation Compositio Math 110 1998 335 367 I Jimenez Calvo J Herranz and G Saez 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 Edita page on the problem by Noam Elkies Retrieved from https en wikipedia org w index php title Hall 27s conjecture amp oldid 1066546371, wikipedia, wiki, book, books, library,
