Fermat's last theorem and Catalan's conjecture in weak exponential arithmetics

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F17%3A00481155" target="_blank" >RIV/67985840:_____/17:00481155 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/17:10369238 RIV/61384399:31140/17:00051179

Výsledek na webu

<a href="http://dx.doi.org/10.1002/malq.201500069" target="_blank" >http://dx.doi.org/10.1002/malq.201500069</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/malq.201500069" target="_blank" >10.1002/malq.201500069</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Fermat's last theorem and Catalan's conjecture in weak exponential arithmetics

Popis výsledku v původním jazyce

We study Fermat's last theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions math formula of models of arithmetical theories (in the language math formula) by a binary (partial or total) function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms math formula. We construct a model math formula and a substructure math formula with e total and math formula (Presburger arithmetic) such that in both math formula and math formula Fermat's last theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples math formula. On the other hand, under the assumption of ABC conjecture (in the standard model), we show that Catalan's conjecture for e is provable in math formula (even in a weaker theory) and thus holds in math formula and math formula. Finally, we also show that Fermat's last theorem for e is provable (again, under the assumption of ABC in math formula) in math formula“coprimality for e''.

Název v anglickém jazyce

Fermat's last theorem and Catalan's conjecture in weak exponential arithmetics

Popis výsledku anglicky

We study Fermat's last theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions math formula of models of arithmetical theories (in the language math formula) by a binary (partial or total) function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms math formula. We construct a model math formula and a substructure math formula with e total and math formula (Presburger arithmetic) such that in both math formula and math formula Fermat's last theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples math formula. On the other hand, under the assumption of ABC conjecture (in the standard model), we show that Catalan's conjecture for e is provable in math formula (even in a weaker theory) and thus holds in math formula and math formula. Finally, we also show that Fermat's last theorem for e is provable (again, under the assumption of ABC in math formula) in math formula“coprimality for e''.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ17-04703Y" target="_blank" >GJ17-04703Y: Kvadratické formy a numerační systémy nad číselnými tělesy</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Mathematical Logic Quarterly

ISSN

0942-5616

e-ISSN

—

Svazek periodika

63

Číslo periodika v rámci svazku

3-4

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

13

Strana od-do

162-174

Kód UT WoS článku

000414581800001

EID výsledku v databázi Scopus

2-s2.0-85033360515