# Smooth solutions to the $abc$ equation: the $xyz$ Conjecture

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

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## Abstract

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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 $H\left(A,B,C\right)=max\left(|A|,|B|,|C|\right)$, and consider primitive solutions ($\mathrm{gcd}\left(A,B,C\right)=1$) having no prime factor larger than ${\left(logH\left(A,B,C\right)\right)}^{\kappa }$, for a given finite $\kappa$. We show that the $abc$ Conjecture implies that for any fixed $\kappa <1$ 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 $\kappa >8$ the $abc$ 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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