Dealing with natural numbers, the following question can be
raised: how close a square and a cube can be? The difference *x*^{3}*-
y ^{2 }*can be exactly zero when

*German Sáez *and *Javier Herranz *of
the "Universitat Politecnica de Catalunya" and I have developed a new algorithm
[5] that
have raised significantly the items of the *good examples of Hall's conjecture
*table. Very briefly, the method starts from the fact that for
*x=t ^{2 }*and

**Table: G***ood examples of Hall's conjecture*

# | x | r | |
---|---|---|---|

1 | 2 | 1.41 | |

2 | 5234 | 4.26 | GPZ |

3 | 8158 | 3.76 | GPZ |

4 | 93844 | 1.03 | GPZ |

5 | 367806 | 2.93 | GPZ |

6 | 421351 | 1.05 | GPZ |

7 | 720114 | 3.77 | GPZ |

8 | 939787 | 3.16 | GPZ |

9 | 28187351 | 4.87 | GPZ |

10 | 110781386 | 1.23 | GPZ |

11 | 154319269 | 1.08 | GPZ |

12 | 384242766 | 1.34 | GPZ |

13 | 390620082 | 1.33 | GPZ |

14 | 3790689201 | 2.20 | GPZ |

15 | 65589428378 | 2.19 | E |

16 | 952764389446 | 1.15 | E |

17 | 12438517260105 | 1.27 | E |

18 | 35495694227489 | 1.15 | E |

19 | 53197086958290 | 1.66 | E |

20 | 5853886516781223 | 46.60 | E |

21 | 12813608766102806 | 1.30 | E |

22 | 23415546067124892 | 1.46 | E |

23 | 38115991067861271 | 6.50 | E |

24 | 322001299796379844 | 1.04 | E |

25 | 471477085999389882 | 1.38 | E |

26 | 810574762403977064 | 4.66 | E |

27 | 9870884617163518770 | 1.90 | JHS |

28 | 42532374580189966073 | 3.47 | JHS |

29 | 51698891432429706382 | 1.75 | JHS |

30 | 44648329463517920535 | 1.79 | JHS |

31 | 231411667627225650649 | 3.71 | JHS |

32 | 601724682280310364065 | 1.88 | JHS |

33 | 4996798823245299750533 | 2.17 | JHS |

34 | 5592930378182848874404 | 1.38 | JHS |

35 | 14038790674256691230847 | 1.27 | JHS |

36 | 77148032713960680268604 | 10.18 | J.B. (JHS) |

37 | 180179004295105849668818 | 5.65 | J.B. (JHS) |

38 | 372193377967238474960883 | 1.33 | JHS |

39 | 664947779818324205678136 | 16.53 | JHS |

40 | 2028871373185892500636155 | 1.14 | J.B. (JHS) |

41 | 10747835083471081268825856 | 1.35 | JHS |

42 | 37223900078734215181946587 | 1.38 | JHS |

43 | 69586951610485633367491417 | 1.22 | AKR |

44 | 3690445383173227306376634720 | 1.51 | JHS |

45 | 162921297743817207342396140787 | 10.66 | AKR |

46 | 1114592308630995805123571151844 | 1.04 | [3] |

47 | 39739590925054773507790363346813 | 3.75 | AKR |

48 | 862611143810724763613366116643858 | 1.10 | AKR |

49 | 1062521751024771376590062279975859 | 1.006 | AKR |

50 | 6078673043126084065007902175846955 | 1.03 | JHS |

GPZ - J. Gebel, A. Pethö and H.G.Zimmer.

E - Noam D. Elkies

JHS - I. Jiménez Calvo, J. Herranz and G. Sáez.

J.B. - Johan Bosman (using the software of JHS)

AKR - S. Aanderaa, L. Kristiansen and H.K. Ruud with an algorithm of their own.
"A Preliminary Report on Search for Good Examples of Hall's Conjecture"
arXiv:1401.4345 ,

A search for values of *x* for which the values of *k* is less than
*16x ^{1/2}* in absolute value was done. The data can be found
here,
in a format readable by PARI, in the form of 704 couples of values

[1] Hall, M.: The Diophantine equation *x ^{3}
- y^{2 }*

[2] Elkies, N.D.: Rational
points near curves and small nonzero | * x^{3} - y^{2}
*| via lattice reduction.

[3] Danilov, L.V.: The Diophantine equation *x ^{3}
- y^{2 }*

[4] Gebel, J., Pethö, A., and Zimmer, H.G.: On Mordell's equation,
*Compositio Math.* **110 **(1998), 335-367.

[5] 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.

Last updated: November 29, 2013