By Arun-Kumar S.

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Thus α satisfies a quadratic equation. Since the ai are all integers, the number [a0 , a1 , . ] = = [a0 , a1 , . . , an , α] 1 a0 + 1 a1 + a2 +···+α can be expressed as a polynomial in α with rational coefficients, so [a0 , a1 , . ] also satisfies a quadratic polynomial. Finally, α ∈ Q because periodic continued fractions have infinitely many terms. 36 CHAPTER 7. 3 The CF expansions of a qudratic irrationals x is purely periodic iff x<0 and −1 ≤ x < 0 P roof : (⇐=) Assume x > 1 xi+1 = 1 xi − ai ; 1 xi+1 x>1 and −1 ≤ = xi − ai as x = [a0 , .

N = 2k (1) Let n is a power of 2 k≥2 1 k k ie even integer φ (n) = φ(2 ) = 2 (1 − 2 ) = 2k−1 (2) n does not happen to be power of 2 then it divisible by an odd prime p , then n = pk m where k≥1 and gcd(pk , m) = 1 By multiplicative nature of phi-function φ (n) = φ(pk m) = φ(pk ) φ (m) = pk−1 (p − 1)φ (m) Hence φ(n) is even because 2 | p − 1 . 7 Let p denote prime integer. 4. EULER S GENERALIZATION OF FERMAT S THEOREM 59 Proof Euler in his landmark result generalized this theorem for any integer ( described in next section ), so proof of this theorem can be obtained as a corollary to next theorem.

K + 1. 1) j | (k + 1)! + j, ∀j ∈ 2, . . 1 pα || n means pα | n but pα+1 | n 45 46 CHAPTER 9. 7 If for prime p and n ≥ 1 pα || n! Clearly n = 0 and n = 1 are trivial cases. Therefore we ∞ β= i=1 n−1 and pβ || (n − 1)! 7) k ✷ We therefore have α = β + k where p || n and hence since n! = n(n − 1)! and from above we have pβ || (n − 1)! therefore pα || n! 8 For all m, n prime p for pα || n! m! 10) 2n! n! 1. ; p | n! 28) ✷ 48 CHAPTER 9. 2 We have 2n n a= log2 4 ≤ 2n But since 2n n and since for j ∈ {1, 2, .

### Algorithmic number theory by Arun-Kumar S.

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