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 t
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 su
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
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 al
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
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)
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, a
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
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 cl
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.
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) an
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 lowes
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
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 +
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 non
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 divi
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 b
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).
Prob
Discuss how you understand the phrase economy of the holocaust in the context of Auschwitz. Explain the role
of Auschwitz I, II and III in the economy of the Third Reich.
According to the Business dic