grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2022 x-ens-maths-d__mp

47 maths questions

Q1.1 Proof Existence Proof View
Let $a$ be a real number in the open interval $]0,1[$. Show that there exists $\lambda > 0$ such that the polynomial $$P(x) = x - \lambda x(x-a)(x-1)$$ satisfies the following two properties:
  1. $P([0,1]) = [0,1]$,
  2. $P$ is increasing on $[0,1]$.
Q1.2 Taylor series Computation of a Limit, Value, or Explicit Formula View
We fix a choice of $\lambda$ such that $P_a(x) = x - \lambda x(x-a)(x-1)$ satisfies $P([0,1])=[0,1]$ and $P$ is increasing on $[0,1]$. Let $\left(P_a^{\circ n}\right)_{n\geq 0}$ be the sequence of polynomials defined recursively by $P_a^{\circ 0}(x) = x$ and $P_a^{\circ n+1}(x) = P_a\left(P_a^{\circ n}(x)\right)$.
Show that $P_a^{\circ n}$ converges uniformly to 1 on every compact subset of $]a,1]$ and uniformly to 0 on every compact subset of $[0,a[$.
Q1.4 Taylor series Deduction or Consequence from Prior Results View
Let $b \in \mathbb{R}$ such that $\cos(b) \in ]0,1[$. Show that the sequence of functions $(f_{b,n})_{n\in\mathbb{N}}$ defined by $$f_{b,n}(t) = P_{\cos(b)}^{\circ n}\left(\cos^2\left(\frac{\pi}{2}t\right)\right)$$ converges uniformly to 1 on every compact subset of $]-\cos(b), \cos(b)[$ and converges uniformly to 0 on every compact subset of $[-1,-\cos(b)[\cup]\cos(b),1]$.
Q2.2 Vectors: Lines & Planes Matrix Norm, Convergence, and Inequality View
Let $v_1$ and $v_2$ be two vectors of $\mathcal{H} = \{v\in V \mid B(v,v)=-1 \text{ and } z_v > 0\}$. Show that $$B(v_1,v_2) \leq -1,$$ with equality if and only if $v_1 = v_2$.
Q2.3 Vectors: Lines & Planes Projection and Orthogonality View
Deduce that if $v\in\mathcal{H}$, then the restriction of $B$ to $v^\perp$ is an inner product.
Q3.1 Vectors: Lines & Planes Subgroup and Normal Subgroup Properties View
We identify $M_3(\mathbb{R})$ with the linear endomorphisms of $V$. Let $G$ be the set of endomorphisms $g$ such that $$B(gu,gv) = B(u,v)$$ for all $u,v\in V$.
Show that $G$ is a group under composition of linear maps.
Q3.2 Groups Subgroup and Normal Subgroup Properties View
Let $G$ be the group of endomorphisms $g$ of $V$ such that $B(gu,gv)=B(u,v)$ for all $u,v\in V$.
Show that for all $g\in G$, we have $g(\mathcal{H})=\mathcal{H}$ or $-g(\mathcal{H})=\mathcal{H}$.
Q3.3 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View
We denote by $G_0$ the subgroup of $G$ formed by elements $g$ such that $g(\mathcal{H})=\mathcal{H}$. For all $w\in V$ such that $B(w,w)>0$, we define the linear map $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that $s_w^2 = \mathrm{Id}_V$, and determine the eigenvalues and eigenspaces of $s_w$.
Q3.4 Groups Subgroup and Normal Subgroup Properties View
For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that $s_w \in G_0$.
Q3.5 Groups Subgroup and Normal Subgroup Properties View
For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that for all $u,v\in\mathcal{H}$, there exists $w\in V$ such that $B(w,w)>0$ and $s_w(u)=v$.
Q4.1 Vectors: Lines & Planes Normal Vector Determination View
We denote by arcch $:[1,+\infty)\rightarrow\mathbb{R}_+$ the inverse of the hyperbolic cosine. Let $v\in\mathcal{H}$. Show that the set $T_v\mathcal{H}$ of vectors tangent to $\mathcal{H}$ at point $v$ is a vector subspace of $V$ and determine this subspace. Deduce that the restriction of $B$ to $T_v\mathcal{H}$ is an inner product.
Q4.2 Parametric integration View
Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$. The hyperbolic length of $\gamma$ is defined by $$\ell(\gamma) = \int_a^b \sqrt{B(\gamma'(t),\gamma'(t))}\,\mathrm{d}t.$$ Show that if $h:[c,d]\rightarrow[a,b]$ is a diffeomorphism, then $\ell(\gamma) = \ell(\gamma\circ h)$.
Q4.3 Parametric integration View
Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$. The hyperbolic length of $\gamma$ is defined by $$\ell(\gamma) = \int_a^b \sqrt{B(\gamma'(t),\gamma'(t))}\,\mathrm{d}t.$$ Let $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$. Show that $$f'(t) \leq \sqrt{f(t)^2-1}\, n(t).$$
Q4.4 Parametric integration View
Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$, with $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$ satisfying $f'(t) \leq \sqrt{f(t)^2-1}\,n(t)$.
Deduce that $$-B(\gamma(a),\gamma(b)) \leq \operatorname{ch}(\ell(\gamma)).$$
Q4.5 Parametric integration View
Let $u$ and $v$ be two points of $\mathcal{H}$. The hyperbolic distance between $u$ and $v$ is defined by $$d(u,v) = \inf_\gamma \ell(\gamma)$$ where the infimum is taken over the set of continuous and piecewise $\mathcal{C}^1$ paths $\gamma:[a,b]\rightarrow\mathcal{H}$ such that $\gamma(a)=u$ and $\gamma(b)=v$.
Show that $d$ is a distance on $\mathcal{H}$, that is,
  • $d(u,v) = d(v,u)$,
  • $d(u,w) \leq d(u,v)+d(v,w)$, and
  • $d(u,v)=0 \Leftrightarrow u=v$
for all $u,v,w\in\mathcal{H}$.
Q4.6 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
Let $d$ be the hyperbolic distance on $\mathcal{H}$ and $G_0$ the subgroup of endomorphisms preserving $B$ and $\mathcal{H}$. Show that $d(gu,gv) = d(u,v)$ for all $g\in G$.
Q4.7 Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
For all $(t,\theta)\in\mathbb{R}_+\times[0,2\pi]$, define $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ Show that $F$ takes values in $\mathcal{H}$ and that $F:\mathbb{R}_+\times[0,2\pi]\rightarrow\mathcal{H}$ is surjective.
Q4.8 Hyperbolic functions View
For all $(t,\theta)\in\mathbb{R}_+\times[0,2\pi]$, define $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ Calculate, for all $\theta\in[0,2\pi]$, the hyperbolic length of the path $$\begin{array}{rcl} \gamma:[0,b] & \rightarrow & \mathcal{H} \\ t & \mapsto & F(t,\theta). \end{array}$$
Q5.1 Groups Proof of Set Membership, Containment, or Structural Property View
Recall that $\Gamma$ denotes the subgroup of $G_0$ formed by elements $g$ such that $g(V_\mathbb{Z})=V_\mathbb{Z}$.
Show that for all $v,w\in\mathcal{H}$ and all $R\geq 0$, the set $$\{g\in\Gamma \text{ such that } d(gv,w)\leq R\}$$ is finite.
Q5.2 Groups Matrix Group and Subgroup Structure View
We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Verify that $s_{w_1}$, $s_{w_2}$ and $s_{w_3}$ belong to $\Gamma$ and calculate the corresponding matrices.
Q5.3 Groups Proof of Set Membership, Containment, or Structural Property View
We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ We denote by $T$ the set of vectors $v\in\mathcal{H}$ such that $B(v,w_i)\geq 0$ for all $i\in\{1,2,3\}$.
Show that $T$ is compact and contains $v_0 = \begin{pmatrix}0\\0\\1\end{pmatrix}$.
Q5.4 Groups Existence Proof View
We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Let $S_{1,2}$ be the subgroup of $\Gamma$ generated by $s_{w_1}$ and $s_{w_2}$. Let $v\in\mathcal{H}$.
Show that there exists $g\in S_{1,2}$ such that $$B(gv,w_1)\geq 0 \quad \text{and} \quad B(gv,w_2)\geq 0.$$
Q5.6 Groups Existence Proof View
Show that for all $v\in\mathcal{H}$, there exists $g\in\Gamma$ such that $gv\in T$.
Q6.2 Groups Proof of Set Membership, Containment, or Structural Property View
For all $s>1$, we denote by $P_k(s)$ the subset of $P_k$ formed by vectors $v$ such that $z_v\leq s$.
Show that $P_k(s)$ is finite.
Q6.3 Groups Bounding or Estimation Proof View
Show that there exists a constant $C>0$ such that for all $v\in\mathcal{H}$, $$|\{g\in\Gamma \text{ such that } gv\in T\}| \leq C.$$
Q6.4 Groups Bounding or Estimation Proof View
For all $R\in\mathbb{R}$, set $$\Gamma(R) = \{g\in\Gamma \text{ such that } d(v_0,gv_0)\leq R\}.$$ Recall that $\Gamma(R)$ is a finite set. Let $D = \sup_{v\in T} d(v_0,v)$.
Show that, for all $s\geq 0$, $$\frac{1}{C}|\Gamma(\operatorname{arcch}(s)-D)|\cdot|P_k\cap T| \leq |P_k(s)| \leq |\Gamma(\operatorname{arcch}(s)+D)|\cdot|P_k\cap T|.$$
Q6.5 Hyperbolic functions Computation of a Limit, Value, or Explicit Formula View
Let $F:[0,2\pi]\times\mathbb{R}_+\rightarrow\mathcal{H}$ be the map defined by $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ For all $(\theta,\alpha)\in[0,2\pi]\times\mathbb{R}_+$ show that $$d\left(F(t,\theta), F\left(t,\theta+\alpha e^{-t}\right)\right) \underset{t\rightarrow+\infty}{\longrightarrow} \operatorname{arcch}\left(1+\frac{\alpha^2}{8}\right)$$ and that the convergence is uniform on every compact subset of $[0,2\pi]\times\mathbb{R}_+$.
Q6.6 Proof Existence Proof View
For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Show that there exists $r>0$ satisfying the following two properties:
  1. for all $g\in\Gamma(n\ln(2))$, there exists $v\in\Delta(n)$ such that $d(gv_0,v)\leq r$,
  2. for all $v\in\Delta(n)$, there exists $g\in\Gamma(n\ln(2))$ such that $d(gv_0,v)\leq r$.
Q6.7 Proof Bounding or Estimation Proof View
For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Fix $r>0$ as in question 6.6. Show that there exists a constant $A\geq 1$ satisfying the following two properties:
  1. for all $g\in\Gamma(n\ln(2))$, $$|\{v\in\Delta(n) \text{ such that } d(gv_0,v)\leq r\}| \leq A,$$
  2. for all $v\in\Delta(n)$, $$|\{g\in\Gamma(n\ln(2)) \text{ such that } d(gv_0,v)\leq r\}| \leq A.$$
Q6.8 Proof Existence Proof View
Show the existence of constants $C_1 > C_2 > 0$ and $R_0 > 0$ such that, for all $R\geq R_0$, $$C_2 e^R \leq |\Gamma(R)| \leq C_1 e^R.$$
Q6.9 Proof Deduction or Consequence from Prior Results View
Deduce the existence of constants $C_1' > C_2' > 0$ and $s_0 > 1$ such that, for all $k\in\mathbb{N}^*$ and all $s\geq s_0$, $$C_2' s\,|P_k\cap T| \leq |P_k(s)| \leq C_1' s\,|P_k\cap T|.$$
Q7.1 Number Theory Modular Arithmetic Computation View
Let $d$ be a nonzero integer and $u$ be an integer. Show that the sum $$\sum_{k\in\mathbb{Z}/d\mathbb{Z}} e^{\frac{2i\pi}{d}ku}$$ equals $d$ if $u\equiv 0\bmod d$ and $0$ otherwise.
Q7.2 Number Theory GCD, LCM, and Coprimality View
Let $n$ be an integer coprime to $d$. Show that the map $$(a,b)\mapsto(na,nb)$$ is a bijection from $S_{\mathrm{prim}}(d)$ to $S_{\mathrm{prim}}(d)$.
Q7.3 Number Theory GCD, LCM, and Coprimality View
Let $d_1$ and $d_2$ be two coprime integers and $m$ and $n$ be two integers such that $md_1+nd_2=1$. Show that the map $$\varphi:\left((a_1,b_1),(a_2,b_2)\right)\mapsto\left(nd_2 a_1+md_1 a_2,\, nd_2 b_1+md_1 b_2\right)$$ is a bijection from $S_{\mathrm{prim}}(d_1)\times S_{\mathrm{prim}}(d_2)$ to $S_{\mathrm{prim}}(d_1 d_2)$.
Q7.4 Number Theory Arithmetic Functions and Multiplicative Number Theory View
Let $d_1$ and $d_2$ be two coprime integers. Show that for all $(u,v)\in\mathbb{Z}^2$, $$L_{\mathrm{prim}}(u,v,d_1 d_2) = L_{\mathrm{prim}}(u,v,d_1)\,L_{\mathrm{prim}}(u,v,d_2).$$
Q7.5 Number Theory Arithmetic Functions and Multiplicative Number Theory View
Let $p$ be a prime number and $\alpha\geq 1$ be an integer. Show that $$L_{\mathrm{prim}}(u,v,p^\alpha) = L(u,v,p^\alpha) - L(u,v,p^{\alpha-1}).$$
Q7.6 Number Theory Congruence Reasoning and Parity Arguments View
Let $p$ be a prime number. Show that there exists $h\in\mathbb{Z}/p\mathbb{Z}$ such that $h^2=-1$ if and only if $p=2$ or $p\equiv 1\bmod 4$.
Q7.7 Number Theory Congruence Reasoning and Parity Arguments View
Assume that $p$ is a prime number congruent to 1 modulo 4. Show that $(a,b)\in S(p)$ if and only if $$b = ha \quad \text{or} \quad b = -ha$$ where $h$ is a solution of $h^2=-1\bmod p$.
Q7.8 Number Theory Congruence Reasoning and Parity Arguments View
Assume that $p$ is a prime number congruent to 1 modulo 4. Let $\alpha\geq 1$. Show that there exists $j\in\mathbb{Z}/p^\alpha\mathbb{Z}$ such that $j^2=-1$.
Q7.9 Number Theory Congruence Reasoning and Parity Arguments View
Assume that $p$ is a prime number congruent to 1 modulo 4, and fix $j\in\mathbb{Z}/p^\alpha\mathbb{Z}$ such that $j^2=-1$. Let $a\in\mathbb{Z}/p^\alpha\mathbb{Z}$ such that $p$ does not divide $a$. Show that $(a,b)\in S(p^\alpha)$ if and only if $$b = ja \quad \text{or} \quad b = -ja.$$
Q7.10 Number Theory Congruence Reasoning and Parity Arguments View
Assume that $p$ is a prime number congruent to 1 modulo 4, and fix $j\in\mathbb{Z}/p^\alpha\mathbb{Z}$ such that $j^2=-1$. Let $a\in\mathbb{Z}/p^\alpha\mathbb{Z}$ and $k\leq\alpha$ be the largest integer such that $p^k$ divides $a$. Show that if $k<\frac{\alpha}{2}$, then $(a,b)\in S(p^\alpha)$ if and only if $$b\equiv\pm ja\bmod p^{\alpha-k}$$ and that if $k\geq\frac{\alpha}{2}$, then $(a,b)\in S(p^\alpha)$ if and only if $p^{\lceil\frac{\alpha}{2}\rceil}$ divides $b$.
Q7.11 Number Theory Combinatorial Number Theory and Counting View
Show that for all $k\geq 1$, we have $$|S_{\mathrm{prim}}(p^{2k})| \geq \frac{1}{2}p^{2k}.$$
Q7.12 Number Theory Modular Arithmetic Computation View
Let $(u,v)\in\mathbb{Z}^2\setminus(0,0)$. Let $p$ be a prime number congruent to 1 modulo 4 and $\alpha\geq 2$. Show that $L(u,v,p^\alpha)=0$ as soon as $p^{\alpha-1}$ does not divide $u^2+v^2$. Deduce that if $\alpha\geq 3$, then $L_{\mathrm{prim}}(u,v,p^\alpha)=0$ as soon as $p^{\alpha-2}$ does not divide $u^2+v^2$.
Q7.13 Number Theory Combinatorial Number Theory and Counting View
Let $(u,v)\in\mathbb{Z}^2\setminus(0,0)$ and $p$ a prime number congruent to 1 modulo 4. Show that if $p$ does not divide $u^2+v^2$, then $$|L_{\mathrm{prim}}(u,v,p)| \leq 2 \quad \text{and} \quad |L_{\mathrm{prim}}(u,v,p^2)| \leq 1.$$
Q7.14 Number Theory Prime Counting and Distribution View
For every integer $d\geq 2$, we denote by $\mathcal{P}(d)$ the set of prime numbers dividing $d$.
Show the inequality $$d \geq |\mathcal{P}(d)|!$$ Deduce that $$|\mathcal{P}(d)| = \underset{d\rightarrow+\infty}{o}(\log(d)).$$
Q7.15 Number Theory Combinatorial Number Theory and Counting View
Let $d$ be an odd integer. Show that $$|S_{\mathrm{prim}}(d^2)| \geq d^2\, 2^{-|\mathcal{P}(d)|}.$$ Deduce that, for all $\varepsilon>0$, we have $$d^{2-\varepsilon} = \underset{d\rightarrow+\infty}{o}\left(S_{\mathrm{prim}}(d^2)\right).$$
Q7.16 Number Theory Combinatorial Number Theory and Counting View
Let $(u,v)\neq(0,0)$. Show the existence of a constant $C$ (depending on $(u,v)$) such that for all $d>0$, $$|L_{\mathrm{prim}}(u,v,d)| \leq C\, 2^{|\mathcal{P}(d)|}.$$ Deduce that, for all $\varepsilon>0$, we have $$|L_{\mathrm{prim}}(u,v,d)| = \underset{d\rightarrow+\infty}{o}\left(d^\varepsilon\right).$$