Unformatted text preview: q is prime if and only if 3 ( q-1) / 2 â‰¡ -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 diï¬€erent 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 Â± an + b p Â¶ = 0 . Exercise 7. 3.3.19: Let h, p be integers with 1 â‰¤ h â‰¤ p , and assume that p is an odd prime. Show p X n =1 Ë† h X m =1 Â± m + n p Â¶ ! 2 = h ( p-h ) ....
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- Spring '08
- Number Theory, Natural number, Prime number, quadratic residue mod