Unformatted text preview: 2 ³³³ p i r divides n , the number of natural numbers a ± n such that p i 1 p i 2 ³³³ p i r divides a is n p i 1 p i 2 ³³³ p i r . Thus we have the formula '.n/ D j A 1 \ A 2 \ ³³³ \ A n j D j U j ² X i j A i j C X i<j j A i \ A j j ² X i<j<k j A i \ A j \ A k j C ³³³ C . ² 1/ s j A 1 \ ³³³ \ A s j D n ² X i n p i C X i<j n p i p j ² X i<j<k n p i p j p k C ³³³ C . ² 1/ s n p 1 p 2 ³³³ p s : The very nice feature of this formula is that the right side factors, '.n/ D n.1 ² 1 p 1 /.1 ² 1 p 2 / ³³³ .1 ² 1 p s /: Proposition 1.9.18. Let n be a natural number with prime factorization n D p k 1 1 ³³³ p k s s . Then (a) '.n/ D n.1 ² 1 p 1 /.1 ² 1 p 2 / ³³³ .1 ² 1 p s /: (b) '.n/ D '.p k 1 1 / ³³³ '.p k s s /:...
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 Fall '08
 EVERAGE
 Algebra, Natural Numbers, Prime number

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