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Date	November Example questions	Marks available	6	Reference code	EXN.3.AHL.TZ0.2
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Show that	Question number	2	Adapted from	N/A

Question

A Gaussian integer is a complex number, $z$ , such that $z = a + b i$ where $a, b \in ℤ$ . In this question, you are asked to investigate certain divisibility properties of Gaussian integers.

Consider two Gaussian integers, $α = 3 + 4 i$ and $β = 1 - 2 i$ , such that $γ = α β$ for some Gaussian integer $γ$ .

Now consider two Gaussian integers, $α = 3 + 4 i$ and $γ = 11 + 2 i$ .

The norm of a complex number $z$ , denoted by $N (z)$ , is defined by $N (z) = {|z|}^{2}$ . For example, if $z = 2 + 3 i$ then $N (2 + 3 i) = 2^{2} + 3^{2} = 13$ .

A Gaussian prime is a Gaussian integer, $z$ , that cannot be expressed in the form $z = α β$ where $α, β$ are Gaussian integers with $N (α), N (β) > 1$ .

The positive integer $2$ is a prime number, however it is not a Gaussian prime.

Let $α, β$ be Gaussian integers.

The result from part (h) provides a way of determining whether a Gaussian integer is a Gaussian prime.

Find $γ$ .

[2]

Determine whether $\frac{γ}{α}$ is a Gaussian integer.

[3]

On an Argand diagram, plot and label all Gaussian integers that have a norm less than $3$ .

[2]

Given that $α = a + b i$ where $a, b \in ℤ$ , show that $N (α) = a^{2} + b^{2}$ .

[1]

By expressing the positive integer $n = c^{2} + d^{2}$ as a product of two Gaussian integers each of norm $c^{2} + d^{2}$ , show that $n$ is not a Gaussian prime.

[3]

Verify that $2$ is not a Gaussian prime.

[2]

Write down another prime number of the form $c^{2} + d^{2}$ that is not a Gaussian prime and express it as a product of two Gaussian integers.

[2]

Show that $N (α β) = N (α) N (β)$ .

[6]

Hence show that $1 + 4 i$ is a Gaussian prime.

[3]

Use proof by contradiction to prove that a prime number, $p$ , that is not of the form $a^{2} + b^{2}$ is a Gaussian prime.

[6]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$(3 + 4 i) (1 - 2 i) = 11 - 2 i$ (M1)A1

[2 marks]

$\frac{γ}{α} = \frac{41}{25} - \frac{38}{25} i$ (M1)A1

(Since $Re \frac{γ}{α} (= \frac{41}{25})$ and/or $Im \frac{γ}{α} (= - \frac{38}{25})$ are not integers)

$\frac{γ}{α}$ is not a Gaussian integer R1

Note: Award R1 for correct conclusion from their answer.

[3 marks]

$\pm 1, \pm i, 0$ plotted and labelled A1

$1 \pm i, - 1 \pm i$ plotted and labelled A1

Note: Award A1A0 if extra points to the above are plotted and labelled.

[2 marks]

$|z| = \sqrt{a^{2} + b^{2}}$ (and as $N (z) = {|z|}^{2}$ ) A1

then $N (α) = a^{2} + b^{2}$ AG

[1 mark]

$c^{2} + d^{2} = (c + d i) (c - d i)$ A1

and $N (c + d i) = N (c - d i) = c^{2} + d^{2}$ R1

$N (c + d i), N (c - d i) > 1$ (since $c, d$ are positive) R1

so $c^{2} + d^{2}$ is not a Gaussian prime, by definition AG

[3 marks]

$2 (= 1^{2} + 1^{2}) = (1 + i) (1 - i)$ (A1)

$N (1 + i) = N (1 - i) = 2$ A1

so $2$ is not a Gaussian prime AG

[2 marks]

For example, $5 (= 1^{2} + 2^{2}) = (1 + 2 i) (1 - 2 i)$ (M1)A1

[2 marks]

METHOD 1

Let $α = m + n i$ and $β = p + q i$

LHS:

$α β = (m p - n q) + (m q + n p) i$ M1

$N (α β) = {(m p - n q)}^{2} + {(m q + n p)}^{2}$ A1

${(m p)}^{2} - 2 m n p q + {(n q)}^{2} + {(m q)}^{2} + 2 m n p q + {(n p)}^{2}$ A1

${(m p)}^{2} + {(n q)}^{2} + {(m q)}^{2} + {(n p)}^{2}$ A1

RHS:

$N (α) N (β) = (m^{2} + n^{2}) (p^{2} + q^{2})$ M1

${(m p)}^{2} + {(m q)}^{2} + {(n p)}^{2} + {(n q)}^{2}$ A1

LHS = RHS and so $N (α β) = N (α) N (β)$ AG

METHOD 2

Let $α = m + n i$ and $β = p + q i$

LHS

$N (α β) = (m^{2} + n^{2}) (p^{2} + q^{2})$ M1

$= (m + n i) (m - n i) (p + q i) (p - q i)$ A1

$= (m + n i) (p + q i) (m - n i) (p - q i)$

$= ((m p - n q) + (m q + n p) i) ((m p - n q) - (m q + n p) i)$ M1A1

$= {(m p - n q)}^{2} + {(m q + n p)}^{2}$ A1

$N = ((m p - n q) + (m q + n p) i)$ A1

$= N (α) N (β)$ (= RHS) AG

[6 marks]

$N (1 + 4 i) = 17$ which is a prime (in $ℤ$ ) R1

if $1 + 4 i = α β$ then $17 = N (α β) = N (α) N (β)$ R1

we cannot have $N (α), N (β) > 1$ R1

Note: Award R1 for stating that $1 + 4 i$ is not the product of Gaussian integers of smaller norm because no such norms divide $17$

so $1 + 4 i$ is a Gaussian prime AG

[3 marks]

Assume $p$ is not a Gaussian prime

$\Rightarrow p = α β$ where $α, β$ are Gaussian integers and $N (α), N (β) > 1$ M1

$\Rightarrow N (p) = N (α) N (β)$ M1

$p^{2} = N (α) N (β)$ A1

It cannot be $N (α) = 1, N (β) = p^{2}$ from definition of Gaussian prime R1

hence $N (α) = p, N (β) = p$ R1

If $α = a + b i$ then $N (α) = a^{2} + b^{2} = p$ which is a contradiction R1

hence a prime number, $p$ , that is not of the form $a^{2} + b^{2}$ is a Gaussian prime AG

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.6—Simple proof

Show 41 related questions

22M.3.AHL.TZ1.1b.iii:
For $n = 4$ , sketch a diagram clearly showing your answer to part (b)(ii).
22M.1.SL.TZ2.3a:
Prove that the sum of these three integers is always divisible by $3$ .
22M.1.SL.TZ2.3b:
Prove that the sum of the squares of these three integers is never divisible by $3$ .
21N.3.AHL.TZ0.1f:
Sketch the graph of $x^{2} - y^{2} = 1$ , stating the coordinates of any axis intercepts and the equation of each asymptote.
EXM.1.SL.TZ0.1b:
Hence prove that the square of any integer can be written in the form $4t$ or $4t + 1$ , where $t \in {\mathbb{Z}^ + }$ .
SPM.1.SL.TZ0.3a:
Show that ${\left( {2n - 1} \right)^2} + {\left( {2n + 1} \right)^2} = 8{n^2} + 2$ , where $n \in \mathbb{Z}$ .
22M.3.AHL.TZ1.1b.i:
Show that $P_{3} (n) + P_{3} (n + 1) \equiv {(n + 1)}^{2}$ .
22M.3.AHL.TZ1.1b.ii:
State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.
22M.3.AHL.TZ1.1c:
Show that $8 P_{3} (n) + 1$ is the square of an odd number for all $n \in ℤ^{+}$ .
22M.3.AHL.TZ2.2b.ii:
Hence show that ${(α - β)}^{2} + {(β - γ)}^{2} + {(γ - α)}^{2} = 2 p^{2} - 6 q$ .
22M.1.SL.TZ1.9a.ii:
By using a suitable substitution for $a$ , show that $1 - 3 \cos 2 x + 3 \cos^{2} 2 x - \cos^{3} 2 x = 8 \sin^{6} x$ .
SPM.1.SL.TZ0.3b:
Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.
18M.1.AHL.TZ2.H_10b.i:
Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
EXN.1.SL.TZ0.4b:
Prove that the sum of the first $n$ terms of this arithmetic sequence is a square number.
EXN.1.SL.TZ0.4a:
Show that $u_{1} = 4$ .
21N.3.AHL.TZ0.1b:
Show that ${(f (t))}^{2} + {(g (t))}^{2} = f (2 t)$ .
21N.3.AHL.TZ0.1c.i:
$f (i u)$ .
21N.3.AHL.TZ0.1g:
The hyperbola with equation $x^{2} - y^{2} = 1$ can be rotated to coincide with the curve defined by $x y = k, k \in ℝ$ .

Find the possible values of $k$ .
22M.3.AHL.TZ1.1a.i:
For triangular numbers, verify that $P_{3} (n) = \frac{n (n + 1)}{2}$ .
22M.3.AHL.TZ2.2b.i:
Show that $p^{2} - 2 q = α^{2} + β^{2} + γ^{2}$ .
22M.3.AHL.TZ2.2a:
By expanding $(x - α) (x - β) (x - γ)$ show that:

$p = - (α + β + γ)$

$q = α β + β γ + γ α$

$r = - α β γ$ .
21N.1.SL.TZ0.6a:
Show that $2 x - 3 - \frac{6}{x - 1} = \frac{2 x^{2} - 5 x - 3}{x - 1}, x \in ℝ, x \neq 1$ .
21N.1.SL.TZ0.6b:
Hence or otherwise, solve the equation $2 \sin 2 θ - 3 - \frac{6}{\sin 2 θ - 1} = 0$ for $0 \leq θ \leq π, θ \neq \frac{π}{4}$ .
21N.2.AHL.TZ0.6b:
The equation $2 x^{2} - 5 x + 1 = 0$ has two real roots, $α$ and $β$ .

Consider the equation $x^{2} + m x + n = 0$ , where $m, n \in ℤ$ and which has roots $\frac{1}{α^{3}}$ and $\frac{1}{β^{3}}$ .
Without solving $2 x^{2} - 5 x + 1 = 0$ , determine the values of $m$ and $n$ .
21N.3.AHL.TZ0.1a:
Verify that $u = f (t)$ satisfies the differential equation $\frac{d^{2} u}{d t^{2}} = u$ .
21N.3.AHL.TZ0.1e:
Show that ${(f (t))}^{2} - {(g (t))}^{2} = {(f (i u))}^{2} - {(g (i u))}^{2}$ .
21N.3.AHL.TZ0.2b.ii:
By substituting $Y = \frac{d y}{d t}$ , show that $Y = B e^{2 t}$ where $B$ is a constant.
21N.3.AHL.TZ0.2c.iii:
Let the two values found in part (c)(ii) be $λ_{1}$ and $λ_{2}$ .

Verify that $y = F e^{λ_{1} t} + G e^{λ_{2} t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.
21N.2.AHL.TZ0.6a:
Prove the identity ${(p + q)}^{3} - 3 p q (p + q) \equiv p^{3} + q^{3}$ .
EXM.1.SL.TZ0.1a:
Explain why any integer can be written in the form $4k$ or $4k + 1$ or $4k + 2$ or $4k + 3$ , where $k \in \mathbb{Z}$ .
21M.1.SL.TZ2.2:
Consider two consecutive positive integers, $n$ and $n + 1$ .

Show that the difference of their squares is equal to the sum of the two integers.
21N.3.AHL.TZ0.1c.ii:
$g (i u)$ .
21N.3.AHL.TZ0.1d:
Hence find, and simplify, an expression for ${(f (i u))}^{2} + {(g (i u))}^{2}$ .
21N.3.AHL.TZ0.2a.i:
By solving the differential equation $\frac{d y}{d t} = y$ , show that $y = A e^{t}$ where $A$ is a constant.
21N.3.AHL.TZ0.2a.ii:
Show that $\frac{d x}{d t} - x = - A e^{t}$ .
21N.3.AHL.TZ0.2a.iii:
Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$ .
21N.3.AHL.TZ0.2b.i:
By differentiating $\frac{d y}{d t} = - x + y$ with respect to $t$ , show that $\frac{d^{2} y}{d t^{2}} = 2 \frac{d y}{d t}$ .
21N.3.AHL.TZ0.2b.iii:
Hence find $y$ as a function of $t$ .
21N.3.AHL.TZ0.2b.iv:
Hence show that $x = - \frac{B}{2} e^{2 t} + C$ , where $C$ is a constant.
21N.3.AHL.TZ0.2c.i:
Show that $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .
21N.3.AHL.TZ0.2c.ii:
Find the two values for $λ$ that satisfy $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .

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Topic 1—Number and algebra » AHL 1.13—Polar and Euler form

Topic 1—Number and algebra

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