20.500.12556/RUL-170468 Mersennova praštevila in dokaz Lucas-Lehmerjevega testa magistrsko delo Mersenne primes and the proof of the Lucas-Lehmer test V magistrskem delu so obravnavana Mersennova praštevila in Lucas-Lehmerjev test. Mersennova praštevila, ki so oblike 2^p-1, kjer je p praštevilo, so tudi največja znana praštevila. Njihovo praštevilskost preverjamo z Lucas-Lehmerjevim testom, ki je podan kot algoritem v obliki rekurzivnega zaporedja. V nalogi dokažemo veljavnost tega testa na dva načina. Najprej samo z uporabo elementarnih tehnik teorije števil, nato pa tudi z uporabo algebrske teorije. Pri tem vpeljemo vse potrebne pojme iz teorije števil, še posebej linearne in kvadratne kongruence. V celoti dokažemo tudi Zakon o kvadratni recipročnosti, ki je ena od ključnih sestavin dokaza veljavnosti Lucas-Lehmerjevega testa. This master's thesis explores Mersenne primes and the Lucas–Lehmer test. Mersenne primes, which take the form 2^p-1, where p is a prime number, are the largest known prime numbers. Their primality is verified using the Lucas–Lehmer test, which is formulated as an algorithm in the form of a recursive sequence. In this thesis, we prove the validity of the test in two distinct ways. First, we employ only elementary techniques from number theory, then we present a second proof using algebraic methods. Along the way, we introduce all the necessary concepts from number theory, with particular emphasis on linear and quadratic congruences. We also provide a complete proof of the Law of Quadratic Reciprocity, which constitutes one of the key components in establishing the correctness of the Lucas–Lehmer test. Matematika Praštevila Mersennova praštevila popolna števila Lucas-Lehmerjev test kongruence kvadratni ostanki Mersenne primes perfect numbers Lucas-Lehmer test congruences quadratic residues true false false A. Zupančič Slovenski jezik Angleški jezik Magistrsko delo/naloga 2025-07-06 08:31:22 2025-07-06 08:31:24 2025-07-09 06:45:49 0000-00-00 00:00:00 2025 0 Ljubljana 0 Ljubljana 45 str. 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 511(043.2) 185067 241846275 40901.pdf 40901.pdf 1 5464736414435A8AC09F6DD71024DD7E e16d097d7cd786ce7824d898a209a6ac53eb115ddd29164baad679343afb1dc9 f0ca71d5-5a30-11f0-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=212474 Pedagoška fakulteta 0 0 0