Mathematics 400c
50)
Homework (due Mar. 31)
A. Hulpke
Determine the quadratic residues and the quadratic nonresidues modulo 19.
51) Determine (without using a ‘Legendre’ function) the following values of the Legendre symbol (all numbers “under the line” may be assumed to be prime):
3342
2
4323
3
5
9
,
,
,
,
,
1237
3343
9923
4327
97
73
52) Determine which of the following congruences have a solution (you do not need to
give a solution and may assume all moduli are prime):
a) x2 ≡ −1 (mod 5987)
b) x2 + 14x − 35 ≡ 0 (mod 337)
53)∗ Suppose that q is a prime number, that q ≡ 1 (mod 4) and that p = 2q + 1 is also
prime. Show that 2 is a primitive root modulo p.
54) a) Show that a primitive root is never a quadratic residue.
b) Determine all primes p, for which every quadratic nonresidue modulo p is also a primitive root.
55) Let a be a quadratic residue modulo p and suppose that p ≡ 3 (mod 4).
a) Show that x = a(p+1)/4 is a solution to the congruence x2 ≡ a (mod p).
(This is an explicit formula to obtain square roots modulo p for p ≡ 3 (mod 4).
b) Find a solution to the congruence x2 ≡ 7 (mod 787), using the method of part a).
Problems marked with a ∗ are bonus problems for extra credit.
Note Homework which was not picked up in class can be picked up at my office. Abandoned homework will be destroyed at the start of the autumn semester.