p| QED AlbertRMeyer March5,2012 lec5M. 11 AlbertRMeyer - - PowerPoint PPT Presentation

Albert R Meyer March 5, 2012

Prime Factorization

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Albert R Meyer March 5, 2012

Every integer > 1 factors uniquely into a weakly decreasing sequence of primes

Fundamental Thm. of Arithmetic

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Albert R Meyer March 5, 2012

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Unique Prime Factorization

Example:

61394323221 = 53·37·37·37·11·11·7·3·3·3

Albert R Meyer March 5, 2012

Prime Divisibility Lemma: p prime and p | ab

implies p|a or p|b

pf: say not(p|a), so gcd(p,a) = 1 so,

sa + tp = 1

p|

• so p|
• QED

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p|

• b

b b

3

Prime Divisibility

Cor :If p is prime, and

p|a1·a2· ··· ·am then p|ai for some i.

pf: by induction on m.

Unique Prime Factorization

Every integer n > 1 has a unique factorization into primes: p1· ··· ·pk = n with p1 ≥ p2 ≥ ··· ≥ pk

Albert R Meyer March 5, 2012

lec 5M.13

Albert R Meyer March 5, 2012

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Unique Prime Factorization

pf: suppose not. choose smallest n >1:

n = p1·p2···pk = q1·q2···qm p1≥p2≥···≥pk q1≥q2≥···≥qm If q1 =p1, then p2···pk = q2···qm is smaller nonunique.

Unique Prime Factorization

pf: suppose not. choose smallest n >1:

n = p1·p2···pk = q1·q2···qm p1≥p2≥···≥pk q1≥q2≥···≥qm So can assume q1 > p1 ≥ pi

Albert R Meyer March 5, 2012

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Albert R Meyer March 5, 2012

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4

Unique Prime Factorization

pf: but q1|n = p1·p2···pk

so q1|pi for some i by Cor, contradicting that q1 > pi QED

Albert R Meyer March 5, 2012

lec 5M.17

5

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