Unformatted text preview: that everybody is on the same page. Exercise 1. For a, b > 2, show that 2 a + 1 is not divisible by 2 b1. Exercise 2. Let n > d > 0 and assume that d divides n . Show that nφ ( n ) > dφ ( d ) . Exercise 3. Let p be a prime and assume that a has order 3 in the multiplicative group Z * p . Show that (1 + a ) has order 6. Exercise 4. Evaluate the Jacobi symbol ( 103 111 ) . Exercise 5. Let p be a prime congruent 1 mod 4 and assume a 2 + b 2 = p where a is odd. Show that ‡ a p · = 1 . Exercise 6. Show that 1 φ ( n ) = 1 n X d  n μ ( d ) 2 φ ( d ) for all positive integers n . Exercise 7. Show that ∑ ∞ n =1 d ( n ) 2 n s = ζ ( n ) 4 ζ (2 s ) ....
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 Spring '08
 Humphreys,G
 Number Theory, Prime number, multiplicative group Z∗

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