MTH 440/540 FALL 2016 HOMEWORK 7
DUE NOVEMBER 21st
(1) (not from the text) State and prove a formula in the spirit of
3
1 if p ≡ 1, 11 (mod 12),
=
−1 if p ≡ 5, 7 (mod 12),
p
for ( −5
p ).
(2) (not from the text) As in exercise 1, state and prove a formula for ( p7 ).
(3) (exercise 4.7) How many natural number x < 213 satisfy the congruence
x2 ≡ 5
(mod 213 − 1)?
You may assume that 213 − 1 is prime.
(4) (not from the text) Suppose p is an odd prime and a, b are integers not divisible by p. Prove that
either exactly one or all three of the integers a, b, ab are quadratic residues.
(5) (not from the text) Suppose p is an odd prime and b is an integer not divisible by p. Prove that
2b
3b
(p − 1)b
b
+
+
+ ··· +
= 0.
p
p
p
p
(6) (not from the text) Suppose p is prime and a, b, c are integers with p not dividing a. Consider the
quadratic congruence ax2 + bx + c ≡ 0 (mod p).
(a) Suppose p = 2. Determine which quadratic congruences have solutions modulo 2 (which values
of a, b, c give a congruence with a solution).
(b) Suppose p is an odd prime and set d = b2 − 4ac. Prove that ax2 + bx + c ≡ 0 (mod p) if and
only if (2ax + b)2 = d. Deduce that the quadratic congruence ax2 + bx + c ≡ 0 (mod p) has
exactly 0, 1, or 2 distinct solutions modulo p according to ( dp ) = −1, p | d, or ( dp ) = 1.
The following exercises marked with a ? are required for students taking the course as MTH 540. These
exercises can be completed for extra credit for students taking the course as MTH 440.
(7) (?, exercise 4.9) In this problem, we will formulate an analog of quadratic reciprocity for a symbol
like ( aq ), but without the restriction that q be a prime. Suppose n is an odd positive integer, which
Qk
we factor as i=1 pei i . We define the Jacobi symbol ( na ) as follows:
e
k a Y
a i
=
n
pi
i=1
(a) Give an example to show that ( na ) need not imply that a is a perfect square.
(b) Suppose a and b are integers.Prove the following:
(i) ( na )( nb ) = ( ab
n ).
(ii) ( −1
n ) ≡ n (mod 4).
(iii) ( n2 ) = 1 if n ≡ ±1 (mod 8) and −1 otherwise.
(iv) Assume that a is positive and odd. Then ( na ) = (−1)
1
a−1 n−1
2
2
( na ).
(8) (not from the text) Prove that there are infinitely many primes of the form 5k − 1. (Hint: Fix a
positive integer n and consider the number m = 5 · (n!)2 − 1. Prove that if p is a prime dividing m,
then p > n and ( p5 ) = 1.)
2