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

Result code in 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>

Alternative codes found

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

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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''.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GJ17-04703Y" target="_blank" >GJ17-04703Y: Quadratic forms and numeration systems over number fields</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2017

Confidentiality

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

Name of the periodical

Mathematical Logic Quarterly

ISSN

0942-5616

e-ISSN

—

Volume of the periodical

63

Issue of the periodical within the volume

3-4

Country of publishing house

DE - GERMANY

Number of pages

13

Pages from-to

162-174

UT code for WoS article

000414581800001

EID of the result in the Scopus database

2-s2.0-85033360515