grandes-ecoles

Papers (191)

2025

centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38

2024

centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26

2023

centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15

2022

centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23

2021

centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44

2020

centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18

2019

centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26

2018

centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24

2017

centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26

2016

centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20

2015

centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22

2014

centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24

2013

centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9

2012

centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18

2011

centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10

2010

centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27

2014 centrale-maths2__pc

26 maths questions

QI.A.1 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View

Let $F$ and $G$ be two supplementary subspaces of $E$ and $s$ the symmetry with respect to $F$ parallel to $G$. a) Show that $F = F_s$ and $G = G_s$. b) Show that $s \circ s = \operatorname{Id}_E$. Deduce that $s$ is an automorphism of $E$. c) Determine the eigenvalues and eigenspaces of $s$. We will discuss according to the subspaces $F$ and $G$.

QI.A.2 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

Let $s$ be an endomorphism of $E$ such that $s \circ s = \operatorname{Id}_E$. We set $F = \operatorname{Ker}(s - \operatorname{Id}_E)$ and $G = \operatorname{Ker}(s + \operatorname{Id}_E)$. a) Show that $F$ and $G$ are two supplementary subspaces of $E$. b) Deduce that $s$ is a symmetry and specify its elements.

QI.B.1 Invariant lines and eigenvalues and vectors Invariant subspaces and stable subspace analysis View

Let $s$ and $t$ be two symmetries of $E$ that anticommute, that is, such that $s \circ t + t \circ s = 0$. a) Prove the equalities $t(F_s) = G_s$ and $t(G_s) = F_s$. b) Deduce that $F_s$ and $G_s$ have the same dimension and that $n$ is even.

QI.C.1 Matrices Linear Transformation and Endomorphism Properties View

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that $$\left\{ \begin{aligned} \forall i & A_i^2 & = I_n \\ \forall i \neq j & A_i A_j + A_j A_i & = 0 \end{aligned} \right.$$ In both cases, $p$ is called the length of the H-system. Show that the length $p$ of an H-system of endomorphisms of $E$ is bounded above by $n^2$.

QI.C.2 Matrices Linear Transformation and Endomorphism Properties View

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that $$\left\{ \begin{aligned} \forall i & A_i^2 & = I_n \\ \forall i \neq j & A_i A_j + A_j A_i & = 0 \end{aligned} \right.$$ In both cases, $p$ is called the length of the H-system. Show that the existence of an H-system $(S_1, \ldots, S_p)$ of $E$ is equivalent to the existence of an H-system of matrices of size $n$. Deduce that the length of an H-system of $E$ depends only on the dimension $n$ of $E$ and not on the space $E$.

QI.C.3 Matrices Linear Transformation and Endomorphism Properties View

QI.D.1 Matrices Linear Transformation and Endomorphism Properties View

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. We assume here that $n$ is even and we set $n = 2m$. We consider:

an H-system $(S_1, \ldots, S_p, T, U)$ of $E$,
the subspace $E_0 = F_T = \operatorname{Ker}(T - \mathrm{Id})$,
for $j \in \llbracket 1, p \rrbracket$, the endomorphism $R_j = \mathrm{i} U \circ S_j$ of $E$.

a) Show that, for all $j \in \llbracket 1, p \rrbracket$, the subspace $E_0$ is stable under $R_j$. b) For $j \in \llbracket 1, p \rrbracket$, let $s_j$ be the endomorphism of $E_0$ induced by $R_j$. Show that $(s_1, \ldots, s_p)$ is an H-system of $E_0$. c) Deduce that $p(2m) \leqslant p(m) + 2$.

QI.D.2 Matrices Linear Transformation and Endomorphism Properties View

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Show that if $n = 2^d m$ with $m$ odd, then $p(n) \leqslant 2d + 1$.

QI.E.1 Matrices Linear Transformation and Endomorphism Properties View

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Let $N = p(n)$ and $(a_1, \ldots, a_N)$ be an H-system of matrices of size $n$, that is, such that $$\forall i, a_i^2 = I_n \quad \text{and} \quad \forall i \neq j, a_i a_j + a_j a_i = 0$$ By considering the following matrices of $\mathcal{M}_{2n}(\mathbb{C})$ written in block form $$A_j = \left( \begin{array}{cc} a_j & 0 \\ 0 & -a_j \end{array} \right) (j \in \llbracket 1, N \rrbracket), \quad A_{N+1} = \left( \begin{array}{cc} 0 & I_n \\ I_n & 0 \end{array} \right), \quad A_{N+2} = \left( \begin{array}{cc} 0 & \mathrm{i} I_n \\ -\mathrm{i} I_n & 0 \end{array} \right)$$ show that $p(2n) \geqslant N + 2$.

QI.E.2 Matrices Linear Transformation and Endomorphism Properties View

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Determine $p(n)$ as a function of the unique integer $d \in \mathbb{N}$ such that $n$ can be written as $n = 2^d m$ with $m$ odd.

QI.E.3 Matrices Linear Transformation and Endomorphism Properties View

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Write, for each of the integers $n = 1, 2, 4$, an H-system of matrices of size $n$ of length $p(n)$.

QII.A.1 Matrices Structured Matrix Characterization View

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. A matrix of the form $M(a, b)$ will be called a quaternion. We will consider in particular the quaternions $e = I_2 = M(1, 0)$, $I = M(0, 1)$, $J = M(\mathrm{i}, 0)$, $K = M(0, -\mathrm{i})$ and we will denote by $\mathbb{H} = \{M(a, b) \mid (a, b) \in \mathbb{C}^2\}$ the subset of $\mathcal{M}_2(\mathbb{C})$ consisting of all quaternions. We equip the set $\mathcal{C} = \mathcal{M}_2(\mathbb{C})$ of complex matrices with two rows and two columns with addition $+$, multiplication $\times$ in the usual sense, and multiplication by a real number denoted $\cdot$. a) Give, without justification, a basis and the dimension of $\mathcal{C}$ over the field $\mathbb{R}$. b) Show that $\mathbb{H}$ is a real vector subspace of $\mathcal{C}$ and that $\{e, I, J, K\}$ is a basis for it over the field $\mathbb{R}$. c) Show that $\mathbb{H}$ is stable under multiplication.

QII.A.2 Groups Subgroup and Normal Subgroup Properties View

QII.A.3 Matrices Matrix Algebra and Product Properties View

QII.B.1 Matrices Matrix Algebra and Product Properties View

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Verify that, for all $q \in \mathbb{H}$, $q^*$ is the transpose of the matrix whose coefficients are the conjugates of the coefficients of $q$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $(qr)^* = r^* q^*$. c) Show that $q^{**} = q$ for all $q \in \mathbb{H}$ and that $q \mapsto q^*$ is an automorphism of the $\mathbb{R}$-vector space $\mathbb{H}$. d) For $q \in \mathbb{H}$, express $qq^*$ in terms of $N(q)$. Deduce the relation valid for all $(q, r) \in \mathbb{H}^2$ $$N(qr) = N(q)N(r)$$

QII.B.2 Matrices Matrix Algebra and Product Properties View

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Let $(x, y, z, t) \in \mathbb{R}^4$ and $q = xe + yI + zJ + tK$. Express the trace of the matrix $q \in \mathcal{M}_2(\mathbb{C})$ in terms of the real number $x$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $qr - rq = q^* r^* - r^* q^*$. c) Let $a, b, c, d$ be quaternions. Establish the relation $(acb^*)d + d^*(acb^*)^* = (acb^*)^* d^* + d(acb^*)$.
Deduce the identity $(N(a) + N(b))(N(c) + N(d)) = N(ac - d^* b) + N(bc^* + da)$.

QIII.A.1 Matrices Linear Transformation and Endomorphism Properties View

Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by $$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$ We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying $$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$ Show the existence of such a bilinear map $B_n$ when $n$ is one of the integers $1, 2, 4$.
For $n = 2$ (respectively 4) one may consider the product of two complex numbers (respectively of two quaternions).

QIII.A.2 Matrices Linear Transformation and Endomorphism Properties View

Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by $$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$ We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying $$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$ Using question II.B.2 show, for $n = 8$, the existence of a bilinear map satisfying the above. We do not ask you to explicitly write down a bilinear map $B_8$, but only to prove its existence.

QIII.B.1 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ Let $(e_1, \ldots, e_n)$ be the canonical basis of $\mathbb{R}^n$ and, for $i \in \llbracket 1, n \rrbracket$, let $u_i$ be the endomorphism of $\mathbb{R}^n$ defined by $$\forall X \in \mathbb{R}^n, \quad u_i(X) = B(X, e_i)$$ The matrix of $u_i$ in the canonical basis of $\mathbb{R}^n$ will be denoted $A_i$. a) Prove that, for all $X \in \mathbb{R}^n$, we have $$\forall Y = (y_1, \ldots, y_n) \in \mathbb{R}^n, \quad \sum_{i,j=1}^n y_i y_j (u_i(X) \mid u_j(X)) = \|X\|^2 \sum_{i=1}^n y_i^2$$ b) Deduce that the endomorphisms $u_i$ satisfy the relations $$\forall i, j = 1, \ldots, n, \forall X \in \mathbb{R}^n, \quad \|u_i(X)\| = \|X\| \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X)) = 0$$ and more generally $$\forall i, j = 1, \ldots, n, \forall X, X' \in \mathbb{R}^n, \quad (u_i(X) \mid u_i(X')) = (X \mid X') \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X')) + (u_j(X) \mid u_i(X')) = 0$$ c) Prove that the matrices $A_i$ satisfy the relations $\forall i, j = 1, \ldots, n, \quad {}^t A_i A_i = I_n$ and $i \neq j \Rightarrow {}^t A_i A_j + {}^t A_j A_i = 0$.

QIII.B.2 Matrices Linear Transformation and Endomorphism Properties View

QIII.B.3 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Prove that $n$ is an element of $\{1, 2, 4, 8\}$.

QV.A Matrices Matrix Algebra and Product Properties View

Let $(A, +, \times)$ be a commutative ring. For $p \in \mathbb{N}^*$, we denote by $C_p(A)$ the set of sums of $p$ squares of elements of $A$. Prove that for every ring $A$, the sets $C_p(A)$ are stable under multiplication when $p$ equals $1, 2, 4$ or $8$.
You may use the bilinear forms $B_p$ defined in part III and, if necessary, restrict yourself to the case where the ring $A$ is the ring $\mathbb{Z}$ of integers.

QV.B.1 Groups Subgroup and Normal Subgroup Properties View

We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q \in \mathbb{H}$, $N(q) = x^2 + y^2 + z^2 + t^2$ where $q = xe + yI + zJ + tK$. a) Show that $\mathbb{G}$ is a subgroup of $\mathbb{H}$ for addition and that it is stable under multiplication. b) Show that for every $q \in \mathbb{H}$, there exists $\mu \in \mathbb{G}$ such that $N(q - \mu) \leqslant 1$. c) What is the set of $q \in \mathbb{H}$ such that $\forall \mu \in \mathbb{G}, N(q - \mu) \geqslant 1$?

QV.B.2 Number Theory Lattice Points and Geometric Number Theory View

Let $p$ be an odd prime number. For every integer $r \in \mathbb{Z}$, we denote by $\varphi(r)$ the remainder of the Euclidean division of $r^2$ by $p$. We thus have $0 \leqslant \varphi(r) \leqslant p - 1$ and $r^2 - \varphi(r) \in p\mathbb{Z}$. a) Show that the restriction of $\varphi$ to $\left\{0, \ldots, \frac{p-1}{2}\right\}$ is injective. b) We consider the sets $X = \left\{p - \varphi(r) \left\lvert\, 0 \leqslant r \leqslant \frac{p-1}{2}\right.\right\}$ and $Y = \left\{\varphi(s) + 1 \left\lvert\, 0 \leqslant s \leqslant \frac{p-1}{2}\right.\right\}$.
Show that $X$ and $Y$ are contained in $\{1, \ldots, p\}$ and that their intersection is non-empty. Deduce that there exist $u, v \in \left\{0, \ldots, \frac{p-1}{2}\right\}$ and $m \in \{1, \ldots, p-1\}$ such that $u^2 + v^2 + 1 = mp$.

QV.B.3 Number Theory Quadratic Diophantine Equations and Perfect Squares View

We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q = xe + yI + zJ + tK \in \mathbb{H}$, $q^* = xe - yI - zJ - tK$ and $N(q) = x^2 + y^2 + z^2 + t^2$. We still assume that $p$ is an odd prime number. Justify that there exist $m \in \{1, \ldots, p-1\}$ and $\mu = xe + yI + zJ + tK \in \mathbb{G} \backslash \{0\}$ such that $N(\mu) = mp$. We choose $m$ minimal and assume that $m > 1$. a) Show that if $m$ were even, an even number of the integers $x, y, z, t$ would be odd and lead to a contradiction.
You may write $\left(\frac{x-y}{2}\right)^2 + \left(\frac{x+y}{2}\right)^2 = \frac{x^2 + y^2}{2}$. b) We assume $m$ is odd. Show that there exists $\nu \in \mathbb{G}$ such that $N(\mu - m\nu) < m^2$. c) Prove that $\mu' = \frac{1}{m}\mu(\mu - m\nu)^*$ is in $\mathbb{G} \backslash \{0\}$ and that $N(\mu')$ is a multiple of $p$ strictly less than $mp$. Conclude.

QV.B.4 Number Theory Quadratic Diophantine Equations and Perfect Squares View

Show that every natural integer is a sum of four squares of integers.