homework6 - a p a (mod p )) by induction on a . 6. Show...

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THEORY OF NUMBERS, Math 115 A Homework 6 Due Monday November 19 1. What is the remainder of 7 · 8 · 9 · 15 · 16 · 17 · 23 · 24 · 25 · 43 modulo 11? 2. Show that the periods in the decimal expansion of 1 / 83 and 1 / 1997 have at least 40 and 400 digits, respectively. Try to check the exact length period with Maple. Write, e. g., Digits:=200: evalf(1/83); ”. 3. Find the least positive residue of 2 1 , 000 , 000 modulo 37. 4. Show that if p is a prime, then ( a + b ) p a p + b p (mod p ) (Hint: prove and use that ( p k ) is a multiple of p for every k ∈ { 1 , . . . , p - 1 } ). Then expand ( a + b ) p via the binomial theorem). 5. Using the previous formula, prove Fermat’s little Theorem (in its equivalent form,
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Unformatted text preview: a p a (mod p )) by induction on a . 6. Show that 45 is a pseudo-prime to the bases 17 and 19. 7. Show that every Carmichael number is square-free. More precisely, show that if m = p 2 q then m is not a pseudo-prime for the base a = pq + 1, and that ( a, m ) = 1. For this: (a) First observe that pq 6 (mod m ), but ( pq ) r (mod m ) for every r > 1. (b) Conclude, using the binomial theorem, that a m 1 (mod m ) . Why does this imply m is not a pseudo-prime? (c) Show that ( a, m ) = 1 by nding the inverse of a modulo m . Hint: (1+ pq )(1-pq ) = 1-( pq ) 2 . 1...
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This note was uploaded on 04/08/2008 for the course MATH 115A taught by Professor Santos during the Fall '07 term at UC Davis.

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