The sum of two integer squares

Jedan od klasičnih rezultata elementarne teorije brojeva je Fermatov teorem da se svaki prost broj može prikazati kao zbroj dva cjelobrojna kvadrata. Iako se teorem pripisuje Fermatu, tvrdnju je još ranije iskazao Girard, bez dokaza. Nije utvrđeno ni postojanje Fermatovog dokaza te prvi pouzdani dokaz potječe od Eulera. Kasnije su brojni matematičari objavili različite nove dokaze, a glavni je cilj ovog rada prikazati neke od tih dokaza koji su se pojavili u novije vrijeme. U radu je prvo ponuđena verzija dokaza koja se može smatrati klasičnom i pristupačnom, a nakon toga slijede dokazi za koje je potrebno pristup nadograditi daljnjim pojmovima i idejama . Od modernijih dokaza prikazan je onaj Zagierov, poznat kao ”dokaz u jednoj rečenici“. Taj je inspiriran Heath-Brownovim dokazom pa je izložena i njegova ideja. Nadalje, uvršteno je objašnjenje Zagierova dokaza koje je dao Elsholtz, u želji da široj publici pojednostavi i protumači dokaz, sa svim potrebnim provjerama koje su izostavljene u Zagierovu krajnje sažetom članku. Zanimljiva je i geometrijska interpretacija kojom se vizualno vjetrenjačama dočarava Zagierov pristup. U radu je objašnjeno i kako sam dokaz učiniti konstruktivnim, a zatim i Elsholtzova kratka varijanta dokaza Zagierom. Na kraju rada predstavljen je potpuno drugačiji pristup A. D. Christophera koji koristi teoriju particija.One of the classic results of elementary number theory is Fermat’s theorem that every prime number p ≡ 1(mod 4) can be represented as the sum of two integer squares. Although the theorem is attributed to Fermat, Girard made the claim even earlier, without proof. The existence of Fermat’s proof has not been confirmed, and the first reliable evidence originates from Euler. Later, a number of mathematicians published various new proofs, and the primary objective of this paper is to present some of these proofs that have emerged in recent times. The paper first offers a version of the proof that can be considered classic and accessible, followed by proof for which the approach needs to be upgraded with further concepts and ideas. Of the modern proofs, Zagier’s proof is presented, also known as the ”one-sentence proof“. The proof from Heath-Brown inspired him, so this idea is also shown. Furthermore, an explanation of Zagier’s proof provided by Elsholtz is included, in an aspiration to simplify and interpret the proof to a wider audience, with all the necessary analysis omitted in Zagier’s extremely concise article. A geometric interpretation is also interesting, as it visually evokes Zagier’s approach using polygons with shapes reminiscent of windmills. It is also explained how to make the proof itself constructive, and furthermore Elsholtz’s short version of the proof is presented. At the end of the paper, a completely different approach of A. D. Christopher using partition theory is presented