hw_05 - q is prime if and only if 3 q-1 2 -1 mod q...

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Number Theory, Homework 5th batch due: Wednesday, February 25 All exercises are worth 5 points. Please select five problems since there is a cap of 25 points. If you decide to do more than five problems, please indicate which ones you want graded since otherwise the grader will be free to chose problems at will. Sometimes, only five problems will be assigned, in which case the selection issue is moot. Exercise 1. 3.1.13: Let r be a quadratic residue mod m > 2. Show that: r φ ( m ) / 2 1 mod m. Exercise 2. 3.1.21: Let p be an odd prime. Show that the following are equivalent: (a) Every quadratic nonresidue mod p is a primitive root. (b) p is of the form 2 2 n + 1. Exercise 3. 3.2.13: Show that there are infinitely many primes congruent 1 mod 3 and that there are infinitely many primes congruent 2 mod 3. Exercise 4. 3.2.15: Let q = 4 n + 1 (where n is a positive integer). Show that
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Unformatted text preview: q is prime if and only if 3 ( q-1) / 2 ≡ -1 mod q . Exercise 5. 3.3.13: Let p be an odd prime. Suppose the set of non-zero congruence classes mod p is partitioned into two disjoint subsets S and T such that (a) the product of any two elements from the same set (either S or T ) is in S and (b) the product of any pair chosen from different sets (one factor from S the other from T ) lies in T . Show that S consists of the quadratic residues and that T consists of the quadratic nonresidues. Exercise 6. 3.3.16: Let a, b, p be integers. Assume that p is an odd prime and that a and p are relatively prime. Show p X n =1 ± an + b p ¶ = 0 . Exercise 7. 3.3.19: Let h, p be integers with 1 ≤ h ≤ p , and assume that p is an odd prime. Show p X n =1 ˆ h X m =1 ± m + n p ¶ ! 2 = h ( p-h ) ....
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