##### Currency

- USD
- GBP
- AUD
- SGD
- NZD

##### Deadline

- 1 Day
- 2 Days
- 3 Days
- 4-6 Days
- 7-10 Days
- > 10 Days

##### Pages

250 Words

**Question - **Math 324 Assignment 9 (due November 19)
1. (6) Evaluate the Legendre symbol âŽŸâŽ
âŽžâŽœâŽ
âŽ›
991
667 by the method of Section 11.2
2. (6) If k â‰¥ 3, show that there are exactly 4 solutions mod 2k of the congruence
x2 â‰¡ 1 mod 2k. Hint: Problem 7.5.
3. (6) Find a congruence describing all odd primes for which 13 is a quadratic residue.
4. (4 + 4) Determine the number of solutions mod m of the congruences
a) x2 âˆ’ 3x + 1 â‰¡ 0 mod 1073, and
b) x2 + x + 2 â‰¡ 0 mod 1219.
5. (3 + 5) Show that, for all odd primes p,
a) S:= {a + bi: a, b in Z} is a subring of C with Z âˆ© pS âŠ† pZ, and
b) prove Theorem 11.6 by the method of Section G. Hint: Consider the element 1 + i âˆ
...Read More

**Solution Preview - **) is prime since this is a Legendre symbol. From 667 = 23â‹…29, with
23 (â‰¡ âˆ’ 1 mod 4) and 29 (â‰¡ 1 mod 4) both prime, we have âŽŸâŽ
âŽžâŽœâŽ
âŽ›
991
667 = âŽŸâŽ
âŽžâŽœâŽ
âŽ›
991
23âŽŸâŽ
âŽžâŽœâŽ
âŽ›
991
29 =
âˆ’ âŽŸâŽ
âŽžâŽœâŽ
âŽ›
23
991âŽŸâŽ
âŽžâŽœâŽ
âŽ›
29
991 = âˆ’ âŽŸâŽ
âŽžâŽœâŽ
âŽ›
23
2âŽŸâŽ
âŽžâŽœâŽ
âŽ›
29
5 = (âˆ’ 1)âŽŸ

**Original Question Documents**

Email your assignment/project