By Chowdhury K.C.

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**Example text**

3. Since the ﬁnal value of the ﬁrst component of the triple u is gcd(m, n), the algorithm is correct. 8. 6 via the Extended Euclidean Algorithm. This table shows the value of the two triples u and v and the integer q each time q is computed and also at the end: u0 u1 u2 v0 v1 v2 75 1 0 21 0 1 21 0 1 12 1 −3 12 1 −3 9 −1 4 9 −1 4 3 2 −7 3 2 −7 0 −7 25 q 3 1 1 3 The last line shows that gcd(75, 21) = 3 = 75(2) + 21(−7). 2. Prime Numbers A prime number is an integer greater than 1 divisible only by 1 and itself.

It is not true that if both d and e divide m, then (bd − 1)(be − 1) divides bm − 1. All we can say is that the least common multiple of (bd − 1) and (be − 1) must divide bm − 1. The cyclotomic factor Φm (b) is called the primitive part of bm −1. The other part, (bm − 1)/Φm (b), is called the algebraic part of bm − 1. A prime factor of bm − 1 is called primitive if is does not divide bk − 1 for any 0 < k < m. Otherwise, it is called algebraic. Every algebraic factor must divide the algebraic part of bm − 1.

Cyclotomic Polynomials A nonconstant polynomial f (x) with rational coeﬃcients is called irreducible (over the rational numbers) if it is not the product of two nonconstant polynomials with rational coeﬃcients of lower degree than f (x). Irreducible polynomials are analogous to prime numbers. One can prove that every nonconstant polynomial can be written as the product of irreducible ones. The polynomial xm − 1 has m zeros in the complex numbers, namely, all the m-th roots of unity. If m > 1, then xm − 1 can 48 3.