Unformatted text preview: b we have a ³ ³ b if and only if a 2 ³ ³ b 2 . (b) Show that for all positive integers a , b and c , if c ³ ³ ab then c ³ ³ gcd( a, c ) gcd( b, c ). (c) Show that for all positive integers a and b we have gcd( a, b ) = gcd ( a + b, lcm( a, b ) ) . 5: A Hilbert number is a positive integer of the form n = 1 + 4 k for some integer k ≥ 0. A Hilbert prime is a Hilbert number n> 1 whose only Hilbert number factors are 1 and n . (a) List the ﬁrst 20 Hilbert primes. (b) Show that every Hilbert number greater than 1 is either a Hilbert prime or a product of Hilbert primes. (c) Show that the factorization of a Hilbert number into Hilbert primes is not always unique....
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 Spring '08
 ANDREWCHILDS
 Algebra, Factors, Integers, Prime number, positive integers, 4k, Hilbert, 6 K

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