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Introductory Number Theory. Homework #2
All numbers are integers.
1. The Fibonacci sequence {Fn } is defined recursively as follows: F1 = 1, F2 = 1, and then Fn = Fn−1 + Fn−2
if n ≥ 3. Prove: gcd(Fn , Fn−1 ) = 1for all n ∈ N. Use induction!
2. Prime power factorization. One way of emphasizing the uniqueness of the prime factorization of a number
n ≥ 0 is to order the primes by size, group equals together as powers. One obtains the following theorem.
Theorem. (Prime power factorization). Every natural number n ≥ 2 can be written uniquely in the form
n = pe11 · · · perr ,
where p1 , . . . , pr are primes such that 2 ≤ p1 < · · · < pr and e1 , . . . , er are positive integers. Uniqueness here
means that if one also has
n = q1f1 · · · qsfs ,
where q1 , . . . , qs are primes such that 2 ≤ q1 < · · · < qs and f1 , . . . , fs are positive integers, then
r = s;
pi = qi for i = 1, . . . , r;
ei = fi for i = 1, . . . , r.
I am not asking you to prove this, it is an immediate consequence of the Fundamental Theorem of Arithmetic;
a different way of stating it. But I want you to be aware of it, understand it, because I plan to use it. Here is
what I want you to prove: Prove that a number n ≥ 2 is a square if and only if in its prime power factorization
n = pe11 · · · perr all the exponents e1 , . . . , er are even.
3. A rational number is one that can be written as a quotient of two integers. If r = a/b is a rational number
b ̸= 0, we say it is written in lowest terms or in reduced form iff b > 0 and gcd(a, b) = 1. It is easy to see that
every rational number other then 0 has a unique representation in lowest terms. We will define a function
f : Q → N as follows. If r > 0, and its representation in lowest terms is a/b, then both a, b > 0 and have no
common factor. In this case we define f (r) = 2a 3b . If r < 0, then in lowest terms r = a/b with b > 0, a < 0.
We define f (r) = 2|a| 5b . Finally, we define f (0) = 1. Prove: f is one to one: If f (r) = f (s), then r = s.
4. Prove: If m, n are natural numbers, if gcd(n, m) = 1 and nm = k 2 for some integer k, then both n and m are
squares; that is, there exist integers i, j such that n = i2 , m = j 2 .
5. Let m ≥ 2. We say that a number a is invertible mod m iff there exists b such that ab ≡ 1 (mod m). Prove:
a is invertible mod m if and only if gcd(a, m) = 1.
6. Let p be prime, p ≥ 3.
(a) Prove: For every a ∈ {1, . . . , p − 1} there exists a unique a′ ∈ {1, . . . , p − 1} such that aa′ ≡ 1 (mod p).
(You can quote results from a previous exercise, if necessary. Don’t prove the same thing twice.)
(b) Prove: a = a′ if and only if a = 1 or a = p − 1. Use this to show that
(p − 1)! ≡ 1 · (p − 1) ≡ −1(mod p).
(c) Show directly the same result also holds for p = 2, thus you proved Wilson’s Theorem: (p − 1) + 1 ≡ 0
(mod p) if p is prime.
7. Prove that 252632 ≡ 1 (mod 52633), but 52633 is divisible by 7.
Hint: One has 52633 = 7 · 73 · 103.
8. Can there exist a trillion consecutive numbers all of which are composite? That is, does there exists n such
that
n + 1, n + 2, n + 3, . . . , n + 109
are all composite?
9. Solve
x ≡ 1 (mod2)
x ≡ 2 (mod3)
x ≡ 3 (mod5)
x ≡ 4 (mod7)
In other words, find all x solving simultaneously all four congruences.
10. Let n ≥ 2. The sum of all integers in the range {1, n − 1} that are relatively prime with n is
2
nφ(n)
.
2