HOMEWORK: SET 1
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. (11 points.) Prove that the function f : N cfw_0 N cfw_0 dened
by
f (n) = [n 3] [n 2]
is surjective.
Note: For each real number x, [x] is the largest integer less than or equal to x.
Problem 2.
PRACTICE PROBLEMS: SET 6
MATH 437/537: PROF. DRAGOS GHIOCA
1. Problems
Problem 1 . Let 1 , . . . , r R. For each > 0 prove that there exist innitely
many positive integers n with the property that for each i = 1, . . . , r we have either
cfw_ni < or cfw_
PRACTICE PROBLEMS: SET 5
MATH 437/537: PROF. DRAGOS GHIOCA
1. Problems
Problem 1. Let f : N C be a multiplicative function. Prove that the function
F : N C dened by
F(n) =
f (d)
d|n
for each n N is also a multiplicative function.
Problem 2. For each n N,
PRACTICE PROBLEMS: SET 4
MATH 437/537: PROF. DRAGOS GHIOCA
1. Problems
Problem 1. Let p be a prime number, and let a Z such that the order of a modulo
p is 3. Prove that
a2 + a + 1 0 (mod p).
Problem 2. For each n N we denote by d(n) the number of positiv
PRACTICE PROBLEMS: SET 1
MATH 437/537: PROF. DRAGOS GHIOCA
1. Problems
Problem 1. Let a, b N. Show that if gcd(a, b) = lcm[a, b], then a = b.
Problem 2. Let n, k N with n 2. Prove that (n 1)2 | (nk 1) if and only
if (n 1) | k.
Problem 3. Let a, b, n N \ c
PRACTICE PROBLEMS: SET 2
MATH 437/537: PROF. DRAGOS GHIOCA
1. Problems
Problem 1. Show that there exist arbitrarily large gaps between any two consecutive primes.
Problem 2. Let p be a prime number, and let a, b Z. Prove that
(a + b)p ap + bp
(mod p).
Pro
SOLUTIONS TO HOMEWORK 4
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. Let p be a prime number satisfying p 1 (mod 3), and let a Z. Find
the number of distinct pairs (x, y) modulo p such that
x3 + ay3 0 (mod p).
Solution. Let x, y Z such that x3 + ay3 0 (mo
SOLUTIONS TO HOMEWORK 1
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. Prove that the function f : N cfw_0 N cfw_0 dened by
f (n) = [n 3] [n 2]
is surjective.
Solution. It suces to show the following claims:
(a) f (0) = 0 and f (n) as n .
(b) f (n) 1 f (n +
SOLUTIONS TO HOMEWORK 5
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. Let p be a prime number satisfying p 3 (mod 4). Prove that for
each a Z, there exist x, y Z such that x4 + y4 a (mod p).
Solution. We rst claim that the set
A := cfw_x4
(mod p) : x Z
has
SOLUTIONS TO HOMEWORK 2
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. Prove that 4n + 15n 1 is divisible by 9 for all n N.
Solution. We rst claim that 4n 1 0 (mod 3) for all n N. Indeed, 4 1
(mod 3) and therefore 4n 1n (mod 3) for all n N. In conclusion,
3
SOLUTIONS TO HOMEWORK 3
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. Prove that for each positive rational number
such that
a (m)
=
.
b
(n)
a
b
there exist m, n N
Solution. So, we let a and b in lowest terms. We let
r
a=
pi
i
i=1
and
s
b=
qj j
j=1
be the
SOLUTIONS TO HOMEWORK 6
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. Show that there exist innitely many triples (x, y, z) satisfying the
equation x2 + y 2 + z 2 = 1 where each x, y and z is a nonzero rational number.
Solution. We note one solution x = 1
HOMEWORK: SET 4
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. (10 points.) Let p be a prime number satisfying p 1 (mod 3), and
let a Z. Find the number of distinct pairs (x, y) modulo p such that
x3 + ay 3 0
(mod p).
Problem 2. (15 points.) Find all positi
HOMEWORK: SET 6
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. (6 points.) Show that there exist innitely many triples (x, y, z)
satisfying the equation x2 + y 2 + z 2 = 1 where each x, y and z is a nonzero rational
number.
Problem 2. (12 points.) Show that
HOMEWORK: SET 2
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. (7 points.) Prove that 4n + 15n 1 is divisible by 9 for all n N.
Problem 2. (7 points.) Show that 21 divides 10n 2n 8 if and only if 6 divides
n 1.
2
Problem 3. (7 points) Find all positive inte
HOMEWORK: SET 3
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. (6 points.) Prove that for each positive rational number
exist m, n N such that
(m)
a
=
.
b
(n)
a
b
there
Problem 2. (10 points.) Let n 2 be an integer. Prove that each positive integer
less tha
HOMEWORK: SET 5
MATH 437/537: PROF. DRAGOS GHIOCA
Problem 1. (10 points.) Let p be a prime number satisfying p 3 (mod 4).
Prove that for each a Z, there exist x, y Z such that x4 + y 4 a (mod p).
Problem 2. (15 points) Let a0 , a1 , , an Z such that |a0 |
PRACTICE PROBLEMS: SET 3
MATH 437/537: PROF. DRAGOS GHIOCA
1. Problems
Problem 1. Let d, n N such that d | n. Prove that if d
n, then
d (d) < n (n).
Problem 2. Let p 5 be a prime number. Let a, b N such that gcd(a, b) = 1 and
p1
1
i=1
i
a
= .
b
Prove that