By Arun-Kumar S.

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**Sample text**

P | ab ⇒ p | a or p | b. 2. p | a1 a2 . . ak ⇒ p | ai for some 1 ≤ i ≤ k. 3. p | q1 q2 . . qk ⇒ p = qi for some 1 ≤ i ≤ k, where q1 , q2 , . . , qk are all primes. We are used to considering primes only on natural numbers. Here is another set of primes over a different set. Consider the set of all even numbers Ze . The set Ze has the following properties: • for all a, b, c ∈ Ze , a + (b + c) = (a + b) + c - associativity. • for all a ∈ Ze , there is an element −a ∈ Ze , such that a + 0 = 0 + a = a, and 0 ∈ Ze - identity element.

41 Proof: (Sketch) The proof is based on showing that if gcd(a, b) = 1, then the series: p≡b 1 p a is divergent. If the series is divergent, then indeed there must be infinitely many primes p such that p ≡b a. Note that p ≡b a implies that p = qb + a for some quotient q and 1 ≤ a < b. 1 Let n ≥ 1 throughout. 2n n 1. 2n ≤ 2. n

2 and 2n/3 < p ≤ n, then p 5. p≤n | p≤2n pr(p) . p < 4n . Proof: 1. As 2n − k ≥ 2(n − k) for 0 ≤ k < n, we have 2n ≤ Also as 2n n n+1 2n 2n − 1 ...

SIGNIFICANCE OF CRT admits a simultaneous solution. r Let M = Mi i=1 mi φ(m ) 61 = M mi φ(m ) The integer x = a1 M1 1 + . . + ar Mr r = ai M φ(mi ) but since gcd (Mi , mi ) =1 , we have r i=1 φ(mi ) Mi φ(mi ) ai Mi full-fills our requirements. Hence x ≡mi ≡mi 1 and so x ≡mi ai for each i. This application is one of the usefulness of Euler’s Theorem in Number Theory. 7 Significance of CRT a (a1 , a2 , . . , ar ) b (b1 , b2 , . . , br ) these representation are unique upto M = mi ((a1 ± b1 )modm1 , (a2 ± b2 )modm2 , .

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