elementary number theory and primality tests

... example, √ 2 and i/2 are algebraic. A complex number is said to be transcendental if it is not algebraic. Both e and π are transcendental. It is in general extremely difficult to prove a number transcendental, ... coprime to n even if n is composite. 5.2 Carmichael numbers Definition 5.2 Suppose n is an odd number > 1. Then we say that n is aCarmichael number if n is not a prime,...

Ngày tải lên: 31/03/2014, 16:21

... q = 1 and r = 986. Notice that if a natural number d divides both 2261 and 1275, then d divides their difference 986 and d still divides 1275. On the other hand, if d divides both 1275 and 986, ... write a = bq + r, with 0 ≤ r < b and q ∈ Z. 6 1. Prime Numbers 3. [Finished?] If r = 0, then b | a, so we output b and terminate. 4. [Shift and Repeat] Set a ← b and b ← r, then...

Ngày tải lên: 07/03/2014, 16:20

... LeVeque, Ftrndamentals of Number Theory, Addison-Wesley, 1977. H. Rademacher, Lectures on Elementary Number Theory, Krieger, 1977. K. H. Rosen, Elementary Number Theory and its Applications, ... any number divides both a and b, it must divide rl, and then, since it divides 14 I. Some Topics in Elementary Number Theory 2 Divisibility and the Euclid...

Ngày tải lên: 31/03/2014, 16:20

Ngày tải lên: 31/03/2014, 16:21

... fact. Exercise 3.2. Prove that if d | a and d | b then d | a − b. Exercise 3.3. Prove that if a ∈ Z then the only positive divisor of both a and a +1is1. Elementary Number Theory W. Edwin Clark Department ... Algorithm). If a and b are integers and b>0 then there exist unique integers q and r satisfying the two conditions: a = bq + r and 0 ≤ r<b.(5.1) In this situation...

Ngày tải lên: 31/03/2014, 16:21

Ngày tải lên: 31/03/2014, 16:21

... natural numbers n. Solution: The statement is evident for n = 1, as k 2 − 1 = (k − 1)(k + 1) is divisible by 8 for any odd natural number k because both (k−1) and (k + 1) are divisible by 2 and one ... has exactly 2 n subsets. 33 APS Prove that if n is a natural number, n 5 /5 + n 4 /2 + n 3 /3 − n/30 is always an integer. Elementary Number Theory Notes c  David A. Santos Janua...

Ngày tải lên: 31/03/2014, 16:21

Ngày tải lên: 31/03/2014, 16:22