Hall hypothesis

Hall hypothesis - unsolved for 2015 number-theoretic hypothesis of an upper bound for solutions of the Diophantine Mordell equation $y^{2}=x^{3}+k$ ${\ displaystyle y ^ {2} = x ^ {3} + k}$ ${\ displaystyle y ^ {2} = x ^ {3} + k}$ for a given $k\neq 0$ ${\ displaystyle k \ neq 0}$ $k \ neq 0$ . It has several formulations of different strengths. It was formulated by Hall in 1971.

Content

Wording and clarification

The initial wording is as follows:

There is a constant $C>0$ ${\ displaystyle C> 0}$ such that if $y^{2}=x^{3}+k$ ${\ displaystyle y ^ {2} = x ^ {3} + k}$ for $x,y,k\in \mathbb {Z}$ ${\ displaystyle x, y, k \ in \ mathbb {Z}}$ and $k\neq 0$ ${\ displaystyle k \ neq 0}$ then $|x|\leqslant C|k|^{2}$ ${\ displaystyle | x | \ leqslant C | k | ^ {2}}$ .

From specific solutions of different equations for different $k$ ${\ displaystyle k}$ can get lower bounds for $C$ ${\ displaystyle C}$ . The most powerful example was found by Elkis in 1998:

447884928428402042307918^{2}-5853886516781223^{3}=-1641843

{\ displaystyle 447884928428402042307918 ^ {2} -5853886516781223 ^ {3} = - 1641843}

From it follows the assessment $C>2171$ ${\ displaystyle C> 2171}$ . This makes the hypothesis implausible in such a formulation, although this formulation is not disproved.

Stark and Trotter in 1980 suggested a weakened version of Hall's hypothesis:

For anyone $\epsilon >0$ ${\ displaystyle \ epsilon> 0}$ there is a constant $C(\epsilon )>0$ ${\ displaystyle C (\ epsilon)> 0}$ such that if $y^{2}=x^{3}+k$ ${\ displaystyle y ^ {2} = x ^ {3} + k}$ for $x,y,k\in \mathbb {Z}$ ${\ displaystyle x, y, k \ in \ mathbb {Z}}$ and $k\neq 0$ ${\ displaystyle k \ neq 0}$ then $|x|\leqslant C(\epsilon )|k|^{2+\epsilon }$ ${\ displaystyle | x | \ leqslant C (\ epsilon) | k | ^ {2+ \ epsilon}}$ .

In view of the implausibility of the original version, the Hall hypothesis now calls the Hall hypothesis its weakened $\epsilon$ ${\ displaystyle \ epsilon}$ .

It is proved that the indicator 2 in the assessment cannot be reduced - the hypothesis becomes incorrect for the assessment of the form $|x|\leqslant C|k|^{2-\epsilon }$ ${\ displaystyle | x | \ leqslant C | k | ^ {2- \ epsilon}}$ (Danilov, 1982).

Davenport Theorem - An Analogue of the Hall Conjecture for Polynomials

In 1965, Davenport proved an analogue of the Hall hypothesis for polynomials:

If a $g(t)^{2}=f(t)^{3}+k(t)$ ${\ displaystyle g (t) ^ {2} = f (t) ^ {3} + k (t)}$ where $k(t)\neq \mathrm {const} ,f(t),g(t)\neq 0$ ${\ displaystyle k (t) \ neq \ mathrm {const}, f (t), g (t) \ neq 0}$ then $\deg f(t)\leqslant 2(\deg k(t)-1)$ ${\ displaystyle \ deg f (t) \ leqslant 2 (\ deg k (t) -1)}$ .

This theorem immediately follows from , an analogue of the ABC hypothesis for polynomials: Let $a(t),b(t),c(t)$ ${\ displaystyle a (t), b (t), c (t)}$ Are pairwise mutually simple non-constant polynomials such that $a+b=c$ ${\ displaystyle a + b = c}$ then

\max\{\deg(a),\deg(b),\deg(c)\}\leqslant \deg(\operatorname {rad} (abc))-1.

{\ displaystyle \ max \ {\ deg (a), \ deg (b), \ deg (c) \} \ leqslant \ deg (\ operatorname {rad} (abc)) - 1.}

Here $\mathrm {rad} (f)$ ${\ displaystyle \ mathrm {rad} (f)}$ Is the radical of a polynomial , that is, the product of its various prime factors.

Substitution $c=g^{2}$ ${\ displaystyle c = g ^ {2}}$ , $a=f^{3}$ ${\ displaystyle a = f ^ {3}}$ , $b=k$ ${\ displaystyle b = k}$ gives 2 inequalities:

3\deg f,2\deg g\leqslant \deg f+\deg g+\deg k-1

{\ displaystyle 3 \ deg f, 2 \ deg g \ leqslant \ deg f + \ deg g + \ deg k-1}

,

from which we obtain the theorem.

Relationship to the ABC Hypothesis

Hall's hypothesis follows from the ABC hypothesis . From the ABC hypothesis immediately follows even stronger, the so-called. Hall's radical hypothesis :

For anyone $\epsilon >0$ ${\ displaystyle \ epsilon> 0}$ there is a constant $C(\epsilon )>0$ ${\ displaystyle C (\ epsilon)> 0}$ such that if $y^{2}=x^{3}+k$ ${\ displaystyle y ^ {2} = x ^ {3} + k}$ for $x,y,k\in \mathbb {Z}$ ${\ displaystyle x, y, k \ in \ mathbb {Z}}$ and $k\neq 0$ ${\ displaystyle k \ neq 0}$ , ${\text{НОД}}(x,y)=1$ ${\ displaystyle {\ text {GCD}} (x, y) = 1}$ then $|x|\leqslant C(\epsilon )\mathrm {rad} (k)^{2+\epsilon }$ ${\ displaystyle | x | \ leqslant C (\ epsilon) \ mathrm {rad} (k) ^ {2+ \ epsilon}}$ .

Here $\mathrm {rad} (k)$ ${\ displaystyle \ mathrm {rad} (k)}$ Is the radical of an integer $k$ ${\ displaystyle k}$ .

It turns out that the ABC hypothesis also follows from Hall's radical hypothesis. However, this statement is nontrivial. ^[1] ^[2]

A generalization of the Hall hypothesis to other degrees is the Pillai hypothesis .

Notes

↑ Schmidt, Wolfgang M. Diophantine approximations and Diophantine equations. - 2nd. - Springer-Verlag , 1996. - Vol. 1467. - P. 205–206. - ISBN 3-540-54058-X .
↑ Bombieri, Gubler. Heights in diophantine geometry. - Cambridge University Press, 2006. - Vol. 652. - P. 424-435. - ISBN 0-511-14061-4 .

Literature

Guy, Richard K. Unsolved problems in number theory. - 3rd. - Springer-Verlag , 2004. - ISBN 978-0-387-20860-2 .
Hall, Jr., Marshall. The Diophantine equation x ³ - y ² = k // Computers in Number Theory. - 1971. - P. 173–198. - ISBN 0-12-065750-3 .

Links

A page on the problem by Noam Elkies
Table of good examples of Marshall Hall's conjecture by Ismael Jimenez Calvo.

[1] Schmidt, Wolfgang M. Diophantine approximations and Diophantine equations. - 2nd. - Springer-Verlag , 1996. - Vol. 1467. - P. 205–206. - ISBN 3-540-54058-X .

[2] Bombieri, Gubler. Heights in diophantine geometry. - Cambridge University Press, 2006. - Vol. 652. - P. 424-435. - ISBN 0-511-14061-4 .