Title Prosti brojevi kroz povijest Author Marin Žufić Mentor Franka Miriam Bruckler (mentor)Committee member Franka Miriam Bruckler (predsjednik povjerenstva)Committee member Sanja Varošanec (član povjerenstva)Committee member Mea Bombardelli (član povjerenstva)Granter University of Zagreb Defense date and country 2017-09-27, Croatia Scientific / art field, NATURAL SCIENCES Abstract Prvo poglavlje bavi se antičkim temeljima istraživanja prostih brojeva. Početke razvoja teorije o prostim brojevima nalazimo u staroj Grčkoj. Prvi koji su se više posvetili problematici prostih brojeva bili su pitagorejci. Na njihove rezultate nastavlja se Euklid, koji svojim Elementima ostavlja neizbrisiv trag, između ostalog i u teoriji prostih brojeva. Priču u staroj Grčkoj zaključuje Eratosten sa najstarijim poznatim algoritmom za pronalaženje prostih brojeva, Eratostenovim sitom. Drugo poglavlje bavi se doprinosima u 17. i 18. st. Uz neke iznimke, razvoj teorije brojeva stagnira do početka 17.st. Za temu rada, glavna imena tog doba su de Fermat, s više rezultata vezanim za proste brojeve, od kojih je najpoznatiji mali Fermatov teorem, te i Mersenne, najpoznatiji po Mersenneovim brojevima. Rad spomenutog dvojca utjecao je na Eulera koji je poopćio mali Fermatov teorem te iskazao zakon kvadratnog reciprociteta, kojeg su dokazali Legendre i Gauß. U trećem poglavlju navodimo neke doprinose u 19. stoljeću, konkretno Gaußov osnovni teorem aritmetike te teorem o prostim brojevima. U posljednjem, četvrtom, poglavlju rada ukratko su opisani razvoji vezani za najpoznatije otvorene probleme o prostim brojevima (Riemannova i Goldbachova hipoteza, hipoteza o parovima blizanaca, …), te neke zanimljivosti poput Ulamove spirale.
Abstract (english) The first chapter deals with the early history of prime number studies. It started in ancient Greece. The first ones to devote themselves to the study of prime numbers were the Pythagoreans. Their results were further developed by Euclid, whose Elements had a huge influence on all of mathematics, including prime numbers theory. We conclude our story about ancient Greek contributions to the topic with the oldest known algorithm for finding prime numbers, the Sieve of Eratosthenes. The second chapter deals with the developments in the 17th and 18th centuries. With minor exceptions, the development of the prime numbers theory was not further developed until the beginning of the 17th century. The main names for our topic in this period are de Fermat, with several contributions, of which the best known is his ’little theorem’, and Mersenne, best known for Mersenne primes, appear. The work of those two mathematicians influenced Euler, who generalised Fermat’s little theorem and expressed the law of quadratic reciprocity. Legendre and Gauß finally proved this law. In the third chapter we deal with 19th century contributions, in particular Gauß’s fundamental theorem of arithmetic and the prime number theorem. The final chapter gives a short overview of main developments concerning the best known open problems in prime number theory (Riemann’s and Goldbach’s conjecture, twin prime conjecture, …) and describes some interesting facts about prime numbers, e.g. the Ulam spiral.
Keywords
Euklid
Eratostenovo sito
mali Fermatov teorem
Mersenneovi brojevi
Euler
Gaußov osnovni teorem aritmetike
teorem o prostim brojevima
Riemannova hipoteza
Goldbachova hipoteza
hipoteza o parovima blizanaca
Ulamova spirala
Keywords (english)
Euclid
Sieve of Eratosthenes
Fermat's little theorem
Mersenne primes
Euler
Gauß’s fundamental theorem of arithmetic
prime number theorem
Riemann’s conjecture
Goldbach’s conjecture
twin prime conjecture
Ulam spiral.
Language croatian URN:NBN urn:nbn:hr:217:067737 Study programme Title: Mathematics Education; specializations in: Mathematics Education Type of resource Text File origin Born digital Access conditions Closed access Terms of use Created on 2018-05-22 11:56:38
