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
2024 x-ens-maths-a__mp

43 maths questions

Q1a Invariant lines and eigenvalues and vectors Diagonalize a matrix explicitly View
Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_n(\mathbb{R})$ $$M_x = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right).$$ Show that the matrix $-M_0$ is diagonalizable and determine its eigenvalues and eigenspaces.
Q2 Number Theory Combinatorial Structures on Permutation Matrices/Groups View
Calculate $$\sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma), \quad \sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma) \nu(\sigma) \quad \text{and} \quad \sum_{\sigma \in \mathfrak{S}_n} \frac{\varepsilon(\sigma)}{\nu(\sigma)+1}.$$
Q3 Number Theory Combinatorial Structures on Permutation Matrices/Groups View
Establish that $$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = -1\right\}$$ and deduce the probability that a permutation of $\mathfrak{S}_n$ drawn uniformly at random has a prescribed signature.
Q4 Number Theory Combinatorial Structures on Permutation Matrices/Groups View
For $\sigma \in \mathfrak{S}_n$, specify the condition on $\nu(\sigma)$ for which $\sigma \in \mathfrak{D}_n$. Deduce that $$\operatorname{Card}\left\{\sigma \in \mathfrak{D}_n : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{D}_n : \varepsilon(\sigma) = -1\right\} + (-1)^{n-1}(n-1).$$
Q5a Matrices Linear Transformation and Endomorphism Properties View
Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^m\right)$ and $\left(1, (X-1), \ldots, (X-1)^m\right)$ are bases of $\mathbb{R}_m[X]$.
Q5b Matrices Linear Transformation and Endomorphism Properties View
Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_m[X] & \longrightarrow & \mathbb{R}_m[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^m\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^m\right)$ at the end.
Q5c Matrices Linear System and Inverse Existence View
Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Establish that $M$ is invertible and explicitly determine its inverse.
Q5d Matrices Matrix Entry and Coefficient Identities View
Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_0, \ldots, u_m\right), \left(v_0, \ldots, v_m\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_k = \sum_{\ell=0}^{k} \binom{k}{\ell} v_\ell, \quad \text{then} \quad \forall k \leqslant m, \quad v_k = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_\ell.$$
Q6 Number Theory Arithmetic Functions and Multiplicative Number Theory View
Show that for any non-zero natural integer $n$, $$D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}.$$
Q7a Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_n, \mathscr{P}(\mathfrak{D}_n))$ equipped with the uniform probability. We define a random variable $Y_n$ by $Y_n(\sigma) = \varepsilon(\sigma)$.
Specify the distribution of $Y_n$.
Q7b Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View
For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_n, \mathscr{P}(\mathfrak{D}_n))$ equipped with the uniform probability. We define a random variable $Y_n$ by $Y_n(\sigma) = \varepsilon(\sigma)$.
Calculate, for all $\varepsilon \in \{-1, 1\}$, $\lim_{n \rightarrow +\infty} \mathbb{P}(Y_n = \varepsilon)$.
Q8a Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Specify the distribution of $Z_n$.
Q8b Discrete Probability Distributions Convergence proof and limit determination View
For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Calculate, for any natural integer $k \leqslant n$, $\lim_{n \rightarrow +\infty} \mathbb{P}(Z_n = k)$.
Q8c Measures of Location and Spread Expectation and Variance via Combinatorial Counting View
For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Determine the average number of fixed points of a random permutation and its limit as $n$ tends to $+\infty$.
Q9 Number Theory Expectation and Variance via Combinatorial Counting View
Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_n$, we recall that there exists, up to order, a unique decomposition $\sigma = c_1 c_2 \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^*$ and $c_1, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_1 \leqslant \ell_2 \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_1 + \ell_2 + \cdots + \ell_{\omega(\sigma)} = n$. We thus obtain a map $\omega : \mathfrak{S}_n \rightarrow \mathbb{N}$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Calculate, for $n \in \{2, 3, 4\}$, the quantity $\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \omega(\sigma)$.
Q10 Number Theory Combinatorial Structures on Permutation Matrices/Groups View
Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_n$, we recall that there exists, up to order, a unique decomposition $\sigma = c_1 c_2 \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^*$ and $c_1, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_1 \leqslant \ell_2 \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_1 + \ell_2 + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Specify $s(n,n)$ and $s(n,1)$ then show that, for $2 \leqslant k \leqslant n-1$, we have $$s(n,k) = s(n-1,k-1) + (n-1)s(n-1,k).$$ For $\sigma \in \mathfrak{S}_n$, one may distinguish the cases $\sigma(1) = 1$ and $\sigma(1) \neq 1$.
Q11 Sequences and Series Combinatorial Structures on Permutation Matrices/Groups View
Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Establish that, for any real $x$, $$\prod_{i=0}^{n-1}(x+i) = \sum_{k=1}^{n} s(n,k) x^k.$$
Q12 Measures of Location and Spread Convergence of Expectations or Moments View
Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Prove that $\mathbb{E}\left[X_n\right] \underset{n \rightarrow +\infty}{=} \ln(n) + \gamma + O\left(\frac{1}{n}\right)$.
Q13a Measures of Location and Spread Expectation and Variance via Combinatorial Counting View
Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Show that $$\frac{1}{n!} \sum_{k=1}^{n} k(k-1) s(n,k) = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^2}.$$
Q13b Measures of Location and Spread Expectation and Variance via Combinatorial Counting View
Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Deduce that $$\frac{1}{n!} \sum_{k=1}^{n} k^2 s(n,k) = \mathbb{E}\left[X_n\right] + \left(\sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^2}\right).$$
Q14a Measures of Location and Spread Convergence of Expectations or Moments View
Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Show that $$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \omega(\sigma)^2 \underset{n \rightarrow +\infty}{=} (2\gamma+1)\ln(n) + c + \ln(n)^2 + O\left(\frac{\ln(n)}{n}\right)$$ for a real number $c$ to be determined.
Q14b Measures of Location and Spread Convergence of Expectations or Moments View
Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Show that $$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} (\omega(\sigma) - \ln(n))^2 \underset{n \rightarrow +\infty}{=} \ln(n) + c + O\left(\frac{\ln(n)}{n}\right).$$
Q15 Measures of Location and Spread View
Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Justify that there exists a real number $C > 0$ such that, for any real $\varepsilon > 0$ and any integer $n \geqslant 1$, we have $$\mathbb{P}\left(\left|X_n - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^2 \ln(n)}.$$
Q16 Number Theory Evaluation of a Finite or Infinite Sum View
For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Let $\left(a_n\right)_{n \geqslant 2}$ be a sequence of real numbers. For $t \in \mathbb{R}$, we set $A(t) = \sum_{2 \leqslant k \leqslant t} a_k$. Let $b : [2, +\infty[ \rightarrow \mathbb{R}$ be a function of class $\mathscr{C}^1$. Show that for any integer $n \geqslant 2$, $$\sum_{k=2}^{n} a_k b(k) = A(n)b(n) - \int_2^n b'(t) A(t) \, \mathrm{d}t$$
Q17a Number Theory Prime Counting and Distribution View
The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^n$.
Handle the cases $n \in \{1, 2, 3\}$.
Q17b Proof Prime Counting and Distribution View
The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^n$.
We assume $n \geqslant 4$ and the result is known at rank $k$ for any integer $k$ between 1 and $n-1$. Establish the result at rank $n$ if $n$ is even.
Q17c Proof Prime Counting and Distribution View
The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^n$.
We assume $n \geqslant 4$ and the result is known at rank $k$ for any integer $k$ between 1 and $n-1$. Let $n = 2m+1$ with $m \in \mathbb{N}$. Justify that $\prod_{\substack{m+1 < p < 2m+1 \\ p \text{ prime}}} p$ divides $\binom{2m+1}{m}$ and show that $\binom{2m+1}{m} \leqslant 4^m$.
Q17d Proof Prime Counting and Distribution View
The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^n$.
Conclude.
Q18 Number Theory Divisibility and Divisor Analysis View
Let $n$ be a non-zero natural integer and let $p$ be a prime number. Justify the formula $\nu_p(n!) = \sum_{k=1}^{+\infty} E\left(\frac{n}{p^k}\right)$ and show that $$\frac{n}{p} - 1 < \nu_p(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$
Q19a Indefinite & Definite Integrals Asymptotic Equivalents and Growth Estimates for Sequences/Series View
By comparison with an integral, establish that $$\sum_{k=1}^{n} \ln(k) \underset{n \rightarrow +\infty}{=} n\ln(n) - n + O(\ln(n))$$
Q19b Number Theory Prime Counting and Distribution View
Justify that $n! = \prod_{\substack{p \leqslant n \\ p \text{ prime}}} p^{\nu_p(n!)}$ and deduce that $$n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} - n\ln(4) < \ln(n!) \leqslant n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} + n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p(p-1)}.$$
Q19c Sequences and Series Convergence/Divergence Determination of Numerical Series View
Justify that the series $\sum_{k \geqslant 2} \frac{\ln(k)}{k(k-1)}$ converges.
Q19d Sequences and Series Prime Counting and Distribution View
Conclude that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} \underset{n \rightarrow +\infty}{=} \ln(n) + O(1)$.
Q20a Sequences and Series Prime Counting and Distribution View
We set, for all real $t \geqslant 2$, $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Show, using the result from question 16, that $$\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} = 1 + \ln_2(n) - \ln_2(2) + \frac{R(n)}{\ln(n)} + \int_2^n \frac{R(t)}{t(\ln(t))^2} \, \mathrm{d}t$$
Q20b Sequences and Series Convergence/Divergence Determination of Numerical Series View
We set, for all real $t \geqslant 2$, $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^2}$ is integrable on $[2, +\infty[$.
Q20c Number Theory Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We set, for all real $t \geqslant 2$, $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Establish that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} \underset{n \rightarrow +\infty}{=} \ln_2(n) + c_1 + O\left(\frac{1}{\ln(n)}\right)$, for a real $c_1 \in \mathbb{R}$ to be determined.
Q21a Number Theory Combinatorial Number Theory and Counting View
Let $x$ be a positive real number greater than or equal to 1 and $q \in \mathbb{N}^*$. Justify that the quantity $$\operatorname{Card}\{n \in \mathbb{N} \cap [1,x] : n \equiv 0 \pmod{q}\} - \frac{x}{q}$$ is bounded in absolute value by a real number independent of $x$ and $q$.
Q21b Number Theory Arithmetic Functions and Multiplicative Number Theory View
For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Prove, using a change of order of summation, that $\frac{1}{x} \sum_{n \leqslant x} \omega(n) \underset{x \rightarrow +\infty}{=} \ln_2(x) + O(1)$.
Q22a Number Theory Arithmetic Functions and Multiplicative Number Theory View
For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Show that $$\frac{1}{x} \sum_{n \leqslant x} \left(\omega(n) - \ln_2(x)\right)^2 \underset{x \rightarrow +\infty}{=} \frac{1}{x}\left(\sum_{n \leqslant x} \omega(n)^2\right) - \ln_2(x)^2 + O\left(\ln_2(x)\right)$$
Q22b Number Theory Arithmetic Functions and Multiplicative Number Theory View
For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Show that $$\sum_{n \leqslant x} \omega(n)^2 = \sum_{\substack{p_1 \leqslant x \\ p_1 \text{ prime}}} \sum_{\substack{p_2 \leqslant x \\ p_2 \text{ prime}}} \operatorname{Card}\left\{n \in \mathbb{N}^* : n \leqslant x, p_1 \mid n \text{ and } p_2 \mid n\right\}$$
Q22c Number Theory Prime Counting and Distribution View
For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Show that $$\left(\sum_{\substack{p_1, p_2 \leqslant x \\ p_1 \neq p_2 \text{ prime}}} \operatorname{Card}\left\{n \in \mathbb{N}^* : n \leqslant x, p_1 \mid n \text{ and } p_2 \mid n\right\}\right) - x\ln_2(x)^2 \underset{x \rightarrow +\infty}{=} O\left(x\ln_2(x)\right)$$ One may estimate the cardinality of the set of pairs of prime numbers $(p_1, p_2)$ such that $p_1 p_2 \leqslant x$ when $x$ tends to $+\infty$.
Q22d Number Theory Arithmetic Functions and Multiplicative Number Theory View
For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Conclude that $\frac{1}{x}\left(\sum_{n \leqslant x} \left(\omega(n) - \ln_2(x)\right)^2\right) \underset{x \rightarrow +\infty}{=} O\left(\ln_2(x)\right)$.
Q23 Number Theory Combinatorial Number Theory and Counting View
For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ We set $\mathscr{S} = \left\{n \geqslant 3 : \left|\frac{\omega(n) - \ln_2(n)}{\sqrt{\ln_2(n)}}\right| \geqslant \left(\ln_2(n)\right)^{1/4}\right\}$. Show that $$\lim_{x \rightarrow +\infty} \frac{1}{x} \operatorname{Card}\{n \leqslant x : n \in \mathscr{S}\} = 0$$ One may begin by writing $\operatorname{Card}(\mathscr{S} \cap [1,x]) \underset{x \rightarrow +\infty}{=} \operatorname{Card}(\mathscr{S} \cap [\sqrt{x}, x]) + O(\sqrt{x})$ and note that in the sum on the right-hand side, the difference $\left|\ln_2(n) - \ln_2(x)\right|$ remains bounded.