20.500.12556/RUL-160626 Catalanova domneva magistrsko delo Catalan's conjecture Catalanova domneva pravi, da je $3^2-2^3=1$ edina rešitev enačbe $x^m-y^n=1$ za naravna števila $x$, $y$, $m$, $n$, kjer velja $m,n > 1$. Domnevo je leta 2002 dokončno dokazal Preda Mihailescu. V magistrskem delu predstavimo zgodovino reševanja Catalanove domneve in obravnavamo posamezne ključne rezultate. Tako dokažemo Catalanovo domnevo v primerih, ko je eden od eksponentov sodo število. Nadaljne reševanje Catalanove domneve sloni na rezultatih iz algebraične teorije števil in posebej teorije ciklotomičnih polj. Med drugim dokažemo Casselsov izrek o deljivosti, ki pravi, da če sta neničelni celi števili $x$, $y$ rešitvi enačbe $x^p-y^q=1$, kjer sta $p$ in $q$ lihi praštevili, potem $p$ deli $y$ in $q$ deli $x$. Dokažemo tudi Mihailescujev izrek o deljivosti, ki za isto enačbo pravi, da $q^2$ deli $x$ in $p^2$ deli $y$. Mihailescujev končni dokaz Catalanove domneve je sestavljen iz dveh delov. V magistrski nalogi v celoti predstavimo del, ko $p$ deli $q-1$ in del dokaza drugega dela, ko $p$ ne deli $q-1$, v katerem je uporabljena Rungeva metoda. Catalan's conjecture asserts that $3^2-2^3=1$ is the only solution to the equation $x^m-y^n=1$ for natural numbers $x$, $y$, $m$, $n$, where $m, n > 1$. The conjecture was proven by Preda Mihailescu in 2002. In the thesis, we present the history of solving Catalan's conjecture and examine individual key results. We prove Catalan's conjecture in cases where one of the exponents is an even number. Further resolution of Catalan's conjecture relies on results from algebraic number theory, particularly the theory of cyclotomic fields. Among other things, we prove Cassels' divisibility theorem, which states that if nonzero integers $x$ and $y$ are solutions to the equation $x^p-y^q=1$, where $p$ and $q$ are odd primes, then $p$ divides $y$ and $q$ divides $x$. We also prove Mihailescu's divisibility theorem for the same equation, stating that $q^2$ divides $x$ and $p^2$ divides $y$. Mihailescu's final proof of Catalan's conjecture consists of two parts. In the master's thesis, we fully present the part where $p$ divides $q-1$ and a part of the proof of the second part when $p$ does not divide $q-1$, using Runge's method. Catalanova domneva Casselsov izrek o deljivosti Mihailescujev izrek o deljivosti dvojni Wieferichov par algebraično številsko polje ciklotomično polje grupni kolobar ulomljeni ideal Stickelbergerjev ideal Catalan’s conjecture Cassels’ divisibility theorem Mihailescu’s divisibility theorem double Wieferich pair number field cyclotomic field group ring fractional ideal Stickelberger’s ideal true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2024-09-01 08:15:03 2024-09-01 08:15:08 2024-09-03 10:15:59 0000-00-00 00:00:00 2024 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 511 142311 206168579 12573.pdf 12573.pdf 1 87BCAFAD531418BFBF519D6484D3ECDF 05386fed6bb16d5120e2509df4475e70d14ea44791e7b3bfd4b0bc7a841219e4 4aa5358a-6829-11ef-8f74-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=189370 Fakulteta za matematiko in fiziko 0 0 0