| Abstract | ABC slutnju su prvi put izrekli u 20. stoljeću matematičari Joseph Oesterlé i David Wiliam Masser u Bonnu 1985.g. Oesterlé i Masser su došli do ABC slutnje potaknuti proučavanjem određenih teza o polinomima i proučavanjem Szpirove slutnje. ABC slutnja kaže da za svaki \(\varepsilon > 0\) postoji samo konačno mnogo trojki relativno prostih prirodnih brojeva \(a, b, c\) za koja vrijedi \(a + b = c\) i za koje je produkt prostih faktora od \(a, b, c\) veći od \(c^{1+\varepsilon}\). ABC trojkom nazivamo relativno proste brojeve \(a\), \(b\) za koje vrijedi \(a+b=c\), dok produkt prostih faktora nekog broja nazivamo radikal. Što je manji radikal u usporedbi s \(c\), veća je kvaliteta trojke. Prvu verziju ABC slutnje nazivamo slaba verzija ABC slutnje, te nakon te verzije su nastale još mnoge verzije. ABC slutnja je važna zbog mnogobrojnih generalizacija i značajnih posljedica koje bi njeno dokazivanje uzrokovalo. Kad bi se ABC slutnja dokazala, dokazali bi se i mnogi dosad neriješeni problemi u matematici. Neke od posljedica: Fermatov posljedni teorem, Catalanova slutnja, Fermat-Catalanova slutnja, Rothov teorem, Hallova slutnja te mnoge druge. Neke od generalizacija: ABC slutnja za polinome, za binarne oblike, za \(n\) cijelih brojeva, Bakerova ABC slutnja za cijele brojeve... Za ABC slutnju se još ne zna je li dokazana. Dokaz od 500 stranica objavio je Shinichi Mochizuki iz Sveučilišta Kyoto u Japanu 2012. Shinichi Mochizuki je japanski matematičar rođen 29. ožujka 1969. godine koji se bavi teorijom brojeva. Matematičari su bili uzbuđeni dokazom, ali su se trudili uhvatiti u koštac s Mochizukijevom "Inter-universal Teichmüller teorijom" (IUT), sasvim novim područjem matematike koju je razvio tijekom desetljeća kako bi se riješio problem. |
| Abstract (english) | The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser (1985). It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime and satisfy \(a + b = c\). If rad denotes the product of the distinct prime factors of abc, the conjecture essentially states that rad is usually not much smaller than \(c\). In other words: if \(a\) and \(b\) are composed from large powers of primes, then \(c\) is usually not divisible by large powers of primes. The abc conjecture has a large number of consequences and generalizations. These include both known results (some of which have been proven separately since the conjecture has been stated) and conjectures for which it gives a conditional proof. While an earlier proof of the conjecture would have been more significant in terms of consequences, the abc conjecture itself remains of interest for the other conjectures it would prove, together with its numerous links with deep questions in number theory. Some of consequences are: Fermat last theorem, Catalan’s conjecture, Fermat-Catalan’s conjecture, Roth’s conjecture, Hall’s conjecture and others. In August 2012, Shinichi Mochizuki released a series of four preprints on Inter-universal Teichmuller Theory which is then applied to prove several famous conjectures in number theory, including Szpiro’s conjecture, the hyperbolic Vojta’s conjecture and the abc conjecture. Mochizuki calls the theory on which this proof is based ”inter-universal Teichmüller theory (IUT)”. The theory is radically different from any standard theory and goes well outside the scope of arithmetic geometry. It was developed over two decades with the last four IUT papers occupying the space of over 500 pages and using many of his prior published papers. |
