By Arun-Kumar S.
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Extra resources for Algorithmic number theory
Pki i ) φ(pi+1 φ((pk11 pk22 . . pki i )pi+1 k k i+1 i+1 = φ(pk11 . . piki ) (pi+1 - pi+1 −1 ) Invoking the induction assumption first factor on right hand side becomes k k k i+1 i+1 i+1 φ(pk11 . . pi+1 ) = (pk11 − pk11 −1 ) . . (piki − piki −1 ) (pi+1 - pi+1 −1 ) This serve to complete the induction step, as well as the proof. 6 φ(360) prime factor of 360 = 23 32 5 φ (360) = 360(1 − 21 )(1 − 13 )(1 − 15 ) = 96 for n>2 , φ(n) is an even integer. Proof Consider two cases when n is power of 2 and when n is not power of two .
Note that a multiplicative inverse need not exist for any arbitrary integer a. For example, 2 doesn’t have a multiplicative inverse modulo 4. 3 puts down necessary and sufficient conditions for existence of an inverse. 3 Elements of Zm which have multiplicative inverses are precisely those that are relatively prime to m. Proof: Rewrite the equation ax ≡m 1 as ax − my = 1. 1, this LDE can be solved iff gcd(a, m) = 1. 4 If p is prime, then all elements in Zp except 0 have multiplicative inverses.
Multiplying Eq. 2 . . 19) (p − 1)! ap−1 p−1 a ap ≡p ≡p ≡p Note that when we vary i in the LHS of Eq. 18, we get a different value of j each time. This accounts for the (p − 1)! term in the RHS of subsequent equations. 9 If ap ≡q a and aq ≡p a where p = q are primes, then apq ≡pq a. 4. 27) ✷ 56CHAPTER 10. 1 Introduction In this lecture, we will discuss Euler s Theorem, Generalisation of Fermat Little Theorem and Chinese Remainder Theorem. 2 EULER s PHI-FUNCTION For n ≥ 1, The number φ(n) denote the number of postive integer not exceeding n , that are relatively prime to n.
Algorithmic number theory by Arun-Kumar S.