March+3

Question 1

a) Compute math \left(\frac{14}{101}\right) math

b) Compute math \left(\frac{-32}{101}\right) math

Question 2

If math p=2^k+1 math is prime, show that 3 is not a quadratic residue.

Combine this with the result of question 2 from March 1 to conclude that every number not divisible by p is congruent to a power of 3 modulo p.

Question 3

For what primes p the number 5 is a quadratic residue?

Question 4

Show that any prime number p which is congruent to 1 modulo 6 is a divisor of some number of the form math n^2+3 math