Unformatted text preview: a p â‰¡ a (mod p )) by induction on a . 6. Show that 45 is a pseudoprime to the bases 17 and 19. 7. Show that every Carmichael number is squarefree. More precisely, show that if m = p 2 q then m is not a pseudoprime for the base a = pq + 1, and that ( a, m ) = 1. For this: (a) First observe that pq 6â‰¡ (mod m ), but ( pq ) r â‰¡ (mod m ) for every r > 1. (b) Conclude, using the binomial theorem, that a m â‰¡ 1 (mod m ) . Why does this imply m is not a pseudoprime? (c) Show that ( a, m ) = 1 by ï¬nding the inverse of a modulo m . Hint: (1+ pq )(1pq ) = 1( pq ) 2 . 1...
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 Fall '07
 Santos
 Number Theory, Binomial Theorem, Remainder, Prime number, Fermat, exact length period

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