List 4: Introduction to Number Theory

Professor: Hossein Movasati

Assistants: Marcus Torres, Roberto Villa�or

P1. Find all integral solutions of the equation

x2 + y2 = 5z2.

P2. Find all pairs m,n ∈ Z>0 such that

m2 − n|m+ n2, and n2 −m|n+m2.

(Hint: Show that |m− n| ≤ 1.)

P3. Find all pairs m,n ∈ Z such that m2 + 4n and n2 + 4m are both squares. (Hint: Show that nand m can't be both positive.)

P4. Find all rational points of the following conics:

(a) x2 + 2y2 = 3.

(b) x2 + xy + y2 = 2.

P5. Show that the equation 7x2 − 13y2 = 1 has no integral solutions.

P6. Let ω = e2πi3 . For α = a+ bω ∈ Z[ω] de�ne the norm

N(α) = a2 − ab+ b2.

Show that α ∈ Z[ω]× ⇔ N(α) = 1. Conclude that Z[ω]× = {±1,±ω,±ω2}.

P7. Show that Z[ω] is an Euclidean domain with the norm N(a+ bω) = a2 − ab+ b2, i.e. show that for

every non-zero elements x, y ∈ Z[ω], there exist q, r ∈ Z[ω] such that

x = qy + r with N(r) < N(y).

P8.

(a) Show that −6 + 48i is a multiple of 2 + 4i in Z[i].

(b) Determine the factorization of −6 + 48i into irredicible factors in Z[i].

P9. Show that Z[√3] is an Euclidean domain with the norm N(a+

√3b) = |a2 − 3b2|.

P10. Let p be a prime number. Show that p is an irreducible element of Z[√3] if and only if the

polynomial X2 − 3 is irreducible in Fp[X]. (Hint: Use that Z[√3] is a unique factorization domain, since

it is in fact an Euclidean domain.)

P11. Wich of the following equations have non-trivial integral solutions?

(a) 3x2 − 5y2 + 7z2 = 0.

(b) 7x2 + 11y2 − 19z2 = 0.

(c) 8x2 − 5y2 − 3z2 = 0.

(d) 11x2 − 3y2 − 41z2 = 0.

P12. Show that if d ∈ Z>0 such that 857 = x2+ dy2 for some x, y ∈ Z>0, then there exist a, b ∈ Z≥0 suchthat 857 = a2 + d · 4b.

P13. Using the previous exercise determine for how many d ∈ Z>0 we can write 857 = x2 + dy2 for

some x, y ∈ Z>0.

P14. Find the fundamental solutions to the following Pell's equations:

(a) x2 − 3y2 = 1.

(b) x2 − 6y2 = 1.

(c) x2 − 624y2 = 1.

P15. Show that if x2 − dy2 = n, d > 0 square-free has an integral solution xy 6= 0 it has in�nitely many.

P16. Show the following genralization of Thue's Lemma (see BMST, Chapter 4). Let n a natural

number, and e, f natural numbers such that ef > n and 1 < e, f < n. Then for every a with gcd(a, n) = 1the congruence

ay ≡ ±x(mod n)

has a solution with 0 < x < e, and 0 < y < f . (Hint: Use Minkowski's Theorem.)

P17. Let q 6≡ 1(mod 4) be a prime number such that, ∀p > 2 prime:

p = α2 + qβ2, for some α, β ∈ Q ⇔ p = x2 + qy2, for some x, y ∈ Z.

Show that q = 2, 3, 7 are the only ones satisfying this property. (Hint: See BMST aplication of Thue's

Lemma.)

P18,19,20. The goal of this exercise is to show that every prime number can be written as a sum

of four squares. Let Q := C× C, with the operations ⊕ and ⊗ de�ned by:

(α, β)⊕ (γ, δ) := (α+ γ, β + δ),

(α, β)⊗ (γ, δ) := (αγ − βδ, αδ + βγ).

These operations make Q a non-commutative ring with identity (1, 0) and zero (0, 0). Consider the

following subring of Q:

H :=

{(a+ ib

2,c+ id

2

)∈ Q|a, b, c, d ∈ Z, a ≡ b ≡ c ≡ d(mod 2)

},

and the function ϕ : Q→ R≥0 de�ned by

ϕ(α, β) := αα+ ββ.

(a) Show that ϕ is a multiplicative function, i.e. ∀x, y ∈ Q

ϕ(x⊗ y) = ϕ(x)ϕ(y),

and that ϕ(x) ∈ Z, ∀x ∈ H. Show that every non-zero element of Q has a multiplicative inverse.

More explicitly, verify that

(α, β)−1 =

(α

ϕ(α, β),−β

ϕ(α, β)

).

(b) Show that (H,⊕,⊗) has the following Euclidean property: given two non-zero elements x, y ∈ H,

there exist elements q, r ∈ H such that

y = q ⊗ x⊕ r with ϕ(r) < ϕ(x).

(Hint: Imitate the proof that Z[i] is an Euclidean domain.)

(c) An element x ∈ H is called central if x ⊗ y = y ⊗ x, for every y ∈ H. A central element x ∈ H is

called prime if ∀y, z ∈ Hx|(y ⊗ z)⇒ x|y or x|z.

Let p ∈ Z a prime number. Show that (p, 0) ∈ H is a central but not prime element. (Hint: Use the

fact that there exist integers x, y ∈ Z such that x2 + y2 ≡ −1(mod p).)

(d) An element x ∈ H non-invertible is called irreducible if ∀y, z ∈ H

x = y ⊗ z ⇒ y or z is invertible in H.

Show that if x ∈ H is central and irreducible, then it is prime. (Hint: If x|(y ⊗ z), consider the setI = {a⊗ x+ b⊗ y}a,b∈H , let w ∈ I such that ϕ(w) > 0 is minimum. Using the Euclidean property

show that I = {c⊗w}c∈H . Since x ∈ I, then x = c⊗w for some c ∈ H. Show that if c is invertiblethen x|y, and if w is invertible then x|z.)

(e) Using items (c) and (d) conclude that (p, 0) is not an irreducible element, and show that there exist

a, b, c, d ∈ Z such that

4p = a2 + b2 + c2 + d2.

(f) Let n ∈ N such that 2n is sum of four squares. Show that n is a sum of four squares.

(Hint: Use a2+b2

2 =(a+b2

)2+(a−b2

)2.)

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