S praštevili, ki so "osnovni gradniki" matematike, so se ukvarjali učenjaki vse od antičnihčasov ... more S praštevili, ki so "osnovni gradniki" matematike, so se ukvarjali učenjaki vse od antičnihčasov dalje. Odločitveni problem praštevilo pravi: za dano naravnoštevilo n ugotovi, ali je praštevilo. Gauss je v knjigi Disquisitiones Arithmeticae (1801) zapisal: "Menim, dačast znanosti narekuje, da z vsemi sredstvi iščemo rešitev tako elegantnega in tako razvpitega problema." Leta 2002 je prišlo do nepričakovanega preboja, ko so Agrawal, Kayal in Saxena, trije indijski matematiki, na internetu predstavili svoj deterministični polinomski algoritem za rešitev tega problema, cf. [AKS], [N]. Delo naj predstavi matematične osnove, glej npr. [S], potrebne za razumevanje učinkovitega testiranja praštevilskosti (tako probabilističnega kot determinističnega) ter analizo najboljših algoritmov. Osrednja motivacija je njihova uporaba v kriptografiji, glej npr. [MOV].
Let p be a prime number. We want to see when an element a of Z has a square root (mod p), i.e., i... more Let p be a prime number. We want to see when an element a of Z has a square root (mod p), i.e., in other words, we want to understand when √ a exists in Z p. A very useful notation to express this condition is the Legendre symbol a p whose value is 1 if a has a square root and is −1 in the other case (and is 0 if p divides a). The main result of the present thesis is Gauss's quadratic reciprocity law which implies that if p and q are two odd prime numbers, then: p q • q p = (−1) p−1 2 q−1 2 There are many different proofs of this theorem and we will analyze two of them which use quite different techniques. The thesis is organized as follows: in the first section we introduce some elementary constructions and the Berlekamp algorithm which allows one to compute a factorization of a polynomial in Z p [x]. Berlekamp theorems will turn out to be useful for proving Euler Criterion. Successively we give the two proofs of Gauss's reciprocity law mentioned above. Finally, in the last section, as a non trivial application of the theory developed, we give the Lucas-Lehmer theorem which allows to decide when a Mersenne number M p is prime.
S praštevili, ki so "osnovni gradniki" matematike, so se ukvarjali učenjaki vse od antičnihčasov ... more S praštevili, ki so "osnovni gradniki" matematike, so se ukvarjali učenjaki vse od antičnihčasov dalje. Odločitveni problem praštevilo pravi: za dano naravnoštevilo n ugotovi, ali je praštevilo. Gauss je v knjigi Disquisitiones Arithmeticae (1801) zapisal: "Menim, dačast znanosti narekuje, da z vsemi sredstvi iščemo rešitev tako elegantnega in tako razvpitega problema." Leta 2002 je prišlo do nepričakovanega preboja, ko so Agrawal, Kayal in Saxena, trije indijski matematiki, na internetu predstavili svoj deterministični polinomski algoritem za rešitev tega problema, cf. [AKS], [N]. Delo naj predstavi matematične osnove, glej npr. [S], potrebne za razumevanje učinkovitega testiranja praštevilskosti (tako probabilističnega kot determinističnega) ter analizo najboljših algoritmov. Osrednja motivacija je njihova uporaba v kriptografiji, glej npr. [MOV].
Let p be a prime number. We want to see when an element a of Z has a square root (mod p), i.e., i... more Let p be a prime number. We want to see when an element a of Z has a square root (mod p), i.e., in other words, we want to understand when √ a exists in Z p. A very useful notation to express this condition is the Legendre symbol a p whose value is 1 if a has a square root and is −1 in the other case (and is 0 if p divides a). The main result of the present thesis is Gauss's quadratic reciprocity law which implies that if p and q are two odd prime numbers, then: p q • q p = (−1) p−1 2 q−1 2 There are many different proofs of this theorem and we will analyze two of them which use quite different techniques. The thesis is organized as follows: in the first section we introduce some elementary constructions and the Berlekamp algorithm which allows one to compute a factorization of a polynomial in Z p [x]. Berlekamp theorems will turn out to be useful for proving Euler Criterion. Successively we give the two proofs of Gauss's reciprocity law mentioned above. Finally, in the last section, as a non trivial application of the theory developed, we give the Lucas-Lehmer theorem which allows to decide when a Mersenne number M p is prime.
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