Prime numbers for high school students

Praštevila predstavljajo osnovne gradnike naravnih števil. Kljub svoji preprosti definiciji ta števila že več kot 2500 let ostajajo uganka. Že starogrški matematiki so opazili skrivnosti teh števil. Še nekoliko več zanimanja so med matematiki doživela v 17. in 18. stoletju, v času Fermata, Eulerja, Gaussa in drugih matematikov tistega časa. Danes pa so nepogrešljiv del kriptografije. V prvem delu magistrske naloge je predstavljena zgodovina praštevil, izpostavljene so ključne prelomnice v spoznavanju le-teh ter opisana je njihova vloga v sodobnem času. V drugem delu je predstavljeno, kako podrobno praštevila spoznajo dijaki po učnem načrtu ter več načinov dokazovanja, da je praštevil neskončno. Zapisanih je nekaj izrekov, ki niso več del učnega načrta, znanje le-teh pa služi kot priprava za reševanje tekmovalnih nalog te tematike. Zbrani so podatki, kako pogosto se v zadnjih desetih letih tema obravnava v raziskovalnih nalogah in na kratko povzete glavne matematične ideje v njih. Predstavljeni so podatki kolikokrat se je v zadnjih desetih letih znanje o praštevilih pojavilo na matematičnih tekmovanjih v osnovnih ali srednjih šolah. Rešenih je nekaj tipov tekmovalnih nalog. V zadnjem delu je predstavljenih še nekaj vprašanj in domnev. Nekatera izmed njih že vrsto let ostajajo odprta.Prime numbers represent the basic building blocks of natural numbers. Despite their simple definition, these numbers have remained a mystery for more than 2,500 years. Already ancient Greek mathematicians noticed the secrets of these numbers. Slightly more interest was experienced among mathematicians in the 17th and 18th centuries, during the time of Fermat, Euler, Gauss, and other mathematicians of the time. The first part of the master\u27s thesis contains some history of prime numbers, highlights key milestones in learning about them and describes their role in modern times. The second part presents what students learn about the prime numbers according to the curriculum and describes several ways of proving that there are infinity many prime numbers. Some theorems that are no longer part of the curriculum are presented. They serve as a preparation for solving competitive tasks on this topic. Data is collected on how often in the last ten years the topic was discussed in high school student research papers and whether in the last ten years the knowledge of prime numbers has also appeared in mathematical competitions in primary or secondary schools. The last part of these thesis presents some questions and assumptions, that still remain open