M3P14/M4P14 - Homework 4
Exercise 1.
√
√
3
Find a polynomial with integer
coefficients
that
has
the
number
2
+
3 as one of its roots.
√
Do the same with the number 5 + i.
Exercise 2.
Given a real number α ∈ R, we denote by αn = [a0 , . . . , an ] its n-th convergent.
a) Show that if α is rational, then there exists a positive integer n such that
α = αn .
b) Show that if D is a positive integer then there are at most finitely many pairs of integers
p, q such that q is positive and if we denote
√
√
p− D
p+ D
and
α=
α=
q
q
then α > 1 and −1 < α < 0.
c) Show that if D is a positive integer which is not a square and α =
i.e. there exist m and n such that
ak+n = ak
√
D then an is periodic,
for all k ≥ m.
Exercise 3.
Set r = 71/5 .
a) Prove that r is algebraic but not rational.
b) Find an explicit constant c > 0 such that for all rationals p/q with p, q ∈ Z and q > 0, we
have |r − p/q| > c/q 5 .
Exercise 4. P
2n
Prove that n≥1 2−(2 ) is transcendental.
1
Exercise 5.
Let α ∈ R be an irrational number. Let pn /qn be the n-th convergent of α. Show that
p2k+1
p2k
<α<
q2k
q2k+1
for any integer k > 0.
Exercise 6.
Show that there are infinitely many primes that are congruent to 5 modulo 6.
Exercise 7.
Find all the solutions of the equation
y 2 − 16x2 = 12
with x, y ∈ Z.
Exercise 8.
Find all the solutions of the equation
y 2 − 2x2 = 1
with x, y ∈ N and 0 < y < 100.
2