## A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO |x 3 − y 2 | VALUES

### BibTeX

@MISC{Calvo_anew,

author = {I. Jiménez Calvo and J. Herranz and G. Sáez},

title = {A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO |x 3 − y 2 | VALUES},

year = {}

}

### OpenURL

### Abstract

Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = |x 3 −y 2 | values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in other cases, it seems difficult to achieve small absolute values. For a given k ̸ = 0, (1.1), known as Mordell’s equation, is an elliptic curve and has only finitely many solutions in integers by Siegel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of |k | was found early by M. Hall [2, 7] by means of a parametric family of the form (1.2) f(t) = t 9 (t9 +6t 6 +15t 3 + 12), g(t) = t15 27 + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27

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Citation Context ...any solutions in integers by Siegel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker =-=[1]-=- and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of |k| was found early by M. Hall [2, 7] by means of a... |

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Citation Context ...that |x3 − y2 | >Cexe. Hall’s conjecture is considered to be a particular case of the related and more general ABC conjecture [11, 9] and both seem hard to prove or disprove. S. Mohit and M. R. Murty =-=[10]-=- show that Hall’s conjecture implies that there are infinitely many primes such that ap−1 ̸≡ 1(mod p16 ) for any a. Since, at present, Hall’s conjecture is neither proved nor disproved, it is worthy t... |

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Citation Context ...ue to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c < 1. A bound concerning the minimal growth rate of |k| was found early by M. Hall =-=[2, 7]-=- by means of a parametric family of the form f(t) = t 9 (t9 + 6t 6 + 15t 3 + 12), (1.2) g(t) = t15 27 + t12 + 4t9 + 8t6 3 f3 (t) − g2 (t) = − 3t6 + 14t3 + 27 , 108 + 5t3 + 1 , 2 with t congruent to 3 ... |

2 |
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Citation Context ...gel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker [1] and improved by H. M. Stark =-=[14]-=-, that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of |k| was found early by M. Hall [2, 7] by means of a parametric family of the form (1... |

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Citation Context ... due to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of |k| was found early by M. Hall =-=[2, 7]-=- by means of a parametric family of the form (1.2) f(t) = t 9 (t9 +6t 6 +15t 3 + 12), g(t) = t15 27 + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27 , 108 + 5t3 +1 , 2 with t congruent to 3 mod 6, ... |

1 |
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Citation Context ...xperimental results for x<700, 000, prompted Marshall Hall [7] to conjecture that |k| cannot be less than Cx1/2 ,for some constant C whose tentative value was fixed to be C =1/5. Later, L. V. Danilov =-=[3]-=- found an infinite family derived from the unbounded solutions of quadratic equations supplying values of |k| < 217 √ 2x1/2 . N.D.Elkies[5]revisedand improved the method, reporting the Fermat-Pell fam... |

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Citation Context ... + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27 , 108 + 5t3 +1 , 2 with t congruent to 3 mod 6, which supplies infinitely many cases with |k| <Cx3/5, where C is a positive constant. H. Davenport =-=[4]-=-, pointed out that the degree of f 3 (t) − g2 (t) isalwaysgreater than half of the degree of f(t). This fact and experimental results for x<700, 000, prompted Marshall Hall [7] to conjecture that |k| ... |

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Citation Context ...ker way as follows: For any exponent e<1/2, a constant Ce > 0 exists such that |x3 − y2 | >Cexe. Hall’s conjecture is considered to be a particular case of the related and more general ABC conjecture =-=[11, 9]-=- and both seem hard to prove or disprove. S. Mohit and M. R. Murty [10] show that Hall’s conjecture implies that there are infinitely many primes such that ap−1 ̸≡ 1(mod p16 ) for any a. Since, at pre... |

1 |
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Citation Context ... (4.2) a ≡ (2C) 1/3 (mod b 2 ). Note that the equation is solvable only when gcd(2C, b) = 1 because a and b are co-prime. The cube root may be computed, for example, using the algorithms described in =-=[12]-=-. Before lifting the solution of (4.2) to modulo 2b3 in (2.5), we must analyze the possible values of the variables whose constraints can be summarized as ⎧ ⎨ even, { C rational, gcd(2C, b) =1, b C ra... |

1 |
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Citation Context ... due to A. Baker [1] and improved by H. M. Stark [14], that places the size of ... |

1 |
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Citation Context ...xperimental results for ... |

1 |
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Citation Context ... + ... |

1 |
The Diophantine equation
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Citation Context ... due to A. Baker [1] and improved by H. M. Stark [14], that places the size of ... |

1 |
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(Show Context)
Citation Context ... (4.2) ... |

1 |
The diophantine equation y 2 − k = x 3
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(Show Context)
Citation Context ... + 4t9 + 8t6 3 f3 (t) − g2 (t) = − 3t6 + 14t3 + 27 , 108 + 5t3 + 1 , 2 with t congruent to 3 mod 6, which supplies infinitely many cases with k < Cx 3/5 , where C is a positive constant. H. Davenport =-=[4]-=-, pointed out that the degree of f 3 (t) − g 2 (t) is always greater than the half of the degree of f(t). This fact and experimental results for x < 700.000, prompted Marshall Hall [7] to conjecture t... |

1 |
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Citation Context ...the b ≤ B values, were v is the exponent due to the factorization. Since a simple trial division with a table of primes was used, the exponent v must be under 0.35 in about half of the instances (see =-=[8]-=-). It is not possible to express the complexity as a fixed function of the bound of x. The value of a is O(bθ ) for an variable exponent θ that is 2.5 in average (see the last paragraph in Section 3).... |