Download EXERCISES NUMBER THEORY Chapter 2

EXERCISES NUMBER THEORY
Chapter 2
14.
Let p be a prime number with p > 2. Prove that x4 ≡ −1(mod p) is solvable if
and only if p ≡ 1(mod 8).
15.
Determine the quadratic residue classes modulo 17 and 19.
16.
Compute
17.
Determine, for any of the integers a below, the set of prime numbers p such that a
is a quadratic residue of p:
a = 5; a = 4; a = 3; a = −3.
18.
Let p be a prime number with p > 2. Prove the following: if x2 ≡ d(mod p) is
solvable, then also x2 ≡ d(mod pk ) is solvable.
19.
Check whether the following congruence equations are solvable:
5
19 )
using Gauss’ lemma.
x2 ≡ 114(mod 127);
x2 ≡ 61(mod 93);
x2 ≡ 837(mod 2996);
x2 ≡ 47(mod 101);
9x2 + 12x + 15 ≡ 0(mod 58);
x2 ≡ 47(mod 143);
8x2 ≡ 2x + 3(mod 175).
20.
Prove that there are infinitely many prime numbers p with p ≡ 1(mod 4).
21.
Denote by K the sum of the integers a ∈ {1, 2, . . . , p − 1} such that a is a quadratic
residue of p, and by N the sum of the integers a ∈ {1, 2, . . . , p − 1} such that a is
is odd.
a quadratic non-residue of p. Prove that p ≡ 3(mod 4) ⇐⇒ N −K
p
22.a) Let p = 4k + 1 be a prime number. Show that
p
p
1 2
√
[ p] + [ 2p] + · · · + [ kp] =
(p − 1) .
12
b) Let p = 4k + 3 be a prime number. Show that
p
p
1
√
[ p] + [ 2p] + · · · + [ kp] ≤
(p − 1)(p − 2) .
12
Hint: Count the lattice points (x, y) under the parabola y =
y ≥ 1 and use in b) that N ≥ K.
√
px with 1 ≤ x < 14 p,