By Arun-Kumar S.

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For all a ∈ Ze , there is an element −a ∈ Ze , such that a + 0 = 0 + a = a, and 0 ∈ Ze - identity element. that this set forms an abelian group since it satisfies associativity, has an identity element (0), and for every even number x ∈ Ze , the negation −e is the unique inverse element under the operation +. Therefore, we have a notion of primality over the ring of even numbers. The only primes in Ze are the numbers of the form 2 · (2k + 1), since they have no factorizations over Ze . 3 Fundamental Theorem of Arithmetic Every positive integer n > 1 is a product of prime numbers, and its factorization into primes is unique up to the order of the factors.

1 For n ≥ 1, φ(n) can be characterised as the number of postive integers less than n and relatively prime to it. The function φ is usually called the Euler phi-function after its originator , ( sometimes the totient ), the functional notion φ(n), however, is credited to Gauss. 2 if n is prime then every number less than n is relatively prime to it , ie φ(n) = n − 1. 3 if p is a prime and k > 1 , then φ(pk ) = pk − pk−1 = pk (1 − p1 ) 57 58 CHAPTER 11. EULER’S φ FUNCTION, GENERALISATION OF FLT, CRT if and only if p does not divide n.

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, . . 34) for x ≥ 2, choose n such that 2n ≤ x < 2n + 2 . n ≥ 1 ⇒ 2n ≥ 2 ⇒ 4n ≥ 2n + 2 ⇒ n ≥ π(2n) ≥ 2n + 2 log2 log2 x ≥ 4 logx 4 logx Therefore a= log2 4 2n+2 4 . 38) log2 where n > 1. Let 2n = 2r for r ≥ 3. 1. Hence 22j+1 j π(22j ) 4 ≤ 22j j Clearly this also holds for j = 1. 51) And hence the result. ✷ 50 CHAPTER 9. 1 Diophantine equations are equations with integer coefficients and which admit only integral solutions.