Smooth solutions to the a b c equation: the x y z Conjecture

Jeffrey C. Lagarias[1]; Kannan Soundararajan[2]

  • [1] University of Michigan Department of Mathematics 530 Church Street Ann Arbor, MI 48109-1043, USA
  • [2] Department of Mathematics Stanford University Department of Mathematics Stanford, CA 94305-2025,USA

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 1, page 209-234
  • ISSN: 1246-7405

Abstract

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This paper studies integer solutions to the a b c equation A + B + C = 0 in which none of A , B , C have a large prime factor. We set H ( A , B , C ) = max ( | A | , | B | , | C | ) , and consider primitive solutions ( gcd ( A , B , C ) = 1 ) having no prime factor larger than ( log H ( A , B , C ) ) κ , for a given finite κ . We show that the a b c Conjecture implies that for any fixed κ < 1 the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed κ > 8 the a b c equation has infinitely many primitive solutions. We outline a proof of the latter result.

How to cite

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Lagarias, Jeffrey C., and Soundararajan, Kannan. "Smooth solutions to the $abc$ equation: the $xyz$ Conjecture." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 209-234. <http://eudml.org/doc/219669>.

@article{Lagarias2011,
abstract = {This paper studies integer solutions to the $\{abc\}$ equation $A+B+C=0$ in which none of $A, B, C$ have a large prime factor. We set $H(A, B,C) = \max (|A|, |B|, |C|)$, and consider primitive solutions ($\{\rm gcd\}(A, B, C)=1$) having no prime factor larger than $(\log H(A, B,C))^\{\kappa \}$, for a given finite $\kappa $. We show that the $\{abc\}$ Conjecture implies that for any fixed $\kappa &lt; 1$ the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed $\kappa &gt;8$ the $\{abc\}$ equation has infinitely many primitive solutions. We outline a proof of the latter result.},
affiliation = {University of Michigan Department of Mathematics 530 Church Street Ann Arbor, MI 48109-1043, USA; Department of Mathematics Stanford University Department of Mathematics Stanford, CA 94305-2025,USA},
author = {Lagarias, Jeffrey C., Soundararajan, Kannan},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = { conjecture},
language = {eng},
month = {3},
number = {1},
pages = {209-234},
publisher = {Société Arithmétique de Bordeaux},
title = {Smooth solutions to the $abc$ equation: the $xyz$ Conjecture},
url = {http://eudml.org/doc/219669},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Lagarias, Jeffrey C.
AU - Soundararajan, Kannan
TI - Smooth solutions to the $abc$ equation: the $xyz$ Conjecture
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 209
EP - 234
AB - This paper studies integer solutions to the ${abc}$ equation $A+B+C=0$ in which none of $A, B, C$ have a large prime factor. We set $H(A, B,C) = \max (|A|, |B|, |C|)$, and consider primitive solutions (${\rm gcd}(A, B, C)=1$) having no prime factor larger than $(\log H(A, B,C))^{\kappa }$, for a given finite $\kappa $. We show that the ${abc}$ Conjecture implies that for any fixed $\kappa &lt; 1$ the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed $\kappa &gt;8$ the ${abc}$ equation has infinitely many primitive solutions. We outline a proof of the latter result.
LA - eng
KW - conjecture
UR - http://eudml.org/doc/219669
ER -

References

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