Devi Purnama sari, 1017031021 (2015) KARAKTERISTIK BILANGAN CARMICHAEL. fakultas MIPA, Universitas Lampung.
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Abstract
Berdasarkan teorema Fermat yang mengatakan bahwa jika p prima dan a bukan kelipatan p, maka ap−1 ≡ 1 (mod p), maka teorema ini memberikan kemungkinan cara untuk mendeteksi bilangan prima atau lebih tepatnya bukan bilangan prima, yaitu jika a relative prima dengan n, an – 1 tidak kongruen dengan 1, maka dengan teorema Fermat tersebut n bukan bilangan prima. Banyak bilangan komposit yang dapat dideteksi dengan menggunakan teorema ini. Untuk bilangan bulat a > 1, himpunan F(a) menyatakan himpunan bilangan positif n yang memenuhi an−1 ≡ 1 mod n. Dengan teorema Fermat, F(a) memuat semua bilangan prima yang bukan pembagi dari a. Jika n ∈ F(a), maka gcd(a,n) = 1, hal ini karena gcd(an−1, n) = 1. Jugna karena an ≡ a mod n;kebaliakn pernyataan tersebut benar, yang menyatakan bahwa a and n relative prima. Suatu bilangan komposit n yang termuat di F(a) disebut pseudoprima-a, atau pseudoprima dengan basis a. Suatu bilangan n yang pseudoprima-a untuk semua a yang relative prima dengan n dinamakan bilangan Carmichael . Bilangan dengan bentuk (6m + 1)(12m + 1)(18m + 1) dengan tiga factor semuanya prima merupakan contoh bilangan Carmichael. Kata Kunci : Bilangan Carmichael , bilangan bulat positif , bilangan prima, bilangan komposit, relative prima, pseudoprima. Recall that Fermat’s “little theorem” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 (mod p). This theorem gives a possible way to detect primes, or more exactly, non-primes: if for a certain a coprime to n, an−1 is not congruent to 1 mod n, then, by the theorem, n is not prime. A lot of composite numbers can indeed be detected by this test, but there are some that evade it. For a fixed a > 1, we write F(a) for the set of positive integers n satisfying an−1 ≡ 1 mod n. By Fermat’s theorem, F(a) includes all primes that are not divisors of a. If n ∈ F(a), then gcd(a,n) = 1, since, clearly, gcd(an−1, n) = 1. Also, an ≡ a mod n;the reverse implication is true provided that a and n are coprime. A composite number n belonging to F(a) is called an apseudoprime, or a pseudoprime to the base a. A number n that is apseudoprime for all a coprime to n is called a Carmichael number. Numbers of the form (6m + 1)(12m + 1)(18m + 1) where all three factors are simultaneously prime are the best known examples of Carmichael numbers. Keywords : Carmichael number, positive integer , prime number, composite number, coprime, pseudoprima.
| Item Type: | Other |
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| Subjects: | Q Science (General) > QA Mathematics |
| Divisions: | Fakultas MIPA > Prodi Matematika |
| Depositing User: | 6395895 . Digilib |
| Date Deposited: | 27 Oct 2015 08:34 |
| Last Modified: | 27 Oct 2015 08:34 |
| URI: | http://digilib.unila.ac.id/id/eprint/13978 |
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