UFM Additional Further Pure

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grandes-ecoles 2022 Q7.16 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 $$|L_{\mathrm{prim}}(u,v,d)| = \underset{d\rightarrow+\infty}{o}\left(d^\varepsilon\right).$$
grandes-ecoles 2022 Q12 Algebraic Number Theory and Minimal Polynomials View
Prove that every divisor of a Hurwitz polynomial is a Hurwitz polynomial.
grandes-ecoles 2022 Q16 Divisibility and Divisor Analysis View
Let $(\Omega, \mathscr{A}, P)$ be a probability space. If $N \in \mathbb{N}^*$ and $p$ is a prime number, we denote by $\nu_p(N)$ the $p$-adic valuation of $N$. For $n \in \mathbb{N}^*$, we define the map $$\psi_n : \mathbb{N}^* \longrightarrow \mathbb{N}^*, \quad x \longmapsto \prod_{i=1}^{n} p_i^{\nu_{p_i}(x)}$$ where $(p_i)_{i \in \mathbb{N}^*}$ is the sequence of prime numbers, ordered in increasing order.
Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Show that $$\forall x \in \mathbb{N}^*, \quad P(X = x) = \lim_{n \rightarrow +\infty} P(\psi_n(X) = x).$$
grandes-ecoles 2022 Q16 Divisibility and Divisor Analysis View
If $N \in \mathbb{N}^*$ and $p$ is a prime number, we denote $\nu_p(N)$ the $p$-adic valuation of $N$. For $n \in \mathbb{N}^*$, we define the application $$\psi_n : \mathbb{N}^* \longrightarrow \mathbb{N}^*, \quad x \longmapsto \prod_{i=1}^{n} p_i^{\nu_{p_i}(x)}$$ where $(p_i)_{i \in \mathbb{N}^*}$ is the sequence of prime numbers, ordered in increasing order. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Show that $$\forall x \in \mathbb{N}^*, \quad \mathbf{P}(X = x) = \lim_{n \rightarrow +\infty} \mathbf{P}(\psi_n(X) = x)$$
grandes-ecoles 2022 Q21 GCD, LCM, and Coprimality View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $s, n \in \mathbb{N}^*$ with $2 \leqslant s \leqslant n$. We randomly draw $s$ numbers from $\{1, 2, \ldots, n\}$ and we denote $P_n(s)$ the probability that these numbers are coprime. Show that $$\lim_{n \rightarrow +\infty} P_n(s) = \frac{1}{\zeta(s)}$$ and give the value of $\lim_{n \rightarrow +\infty} P_n(s)$ in the case where $s = 2$, then $s = 4$, and finally $s = 6$.
grandes-ecoles 2022 Q21 GCD, LCM, and Coprimality View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $s, n \in \mathbb{N}^*$ with $2 \leqslant s \leqslant n$. We randomly draw $s$ numbers from $\{1, 2, \ldots, n\}$ and we denote $P_n(s)$ the probability that these numbers are coprime. Show that $$\lim_{n \rightarrow +\infty} P_n(s) = \frac{1}{\zeta(s)}$$ and give the value of $\lim_{n \rightarrow +\infty} P_n(s)$ in the case where $s = 2$, then $s = 4$, and finally $s = 6$.
grandes-ecoles 2023 Q7 Algebraic Structures in Number Theory View
Show that $\mathscr { D } _ { \rho } ( \mathbb { R } )$ is an integral domain.
grandes-ecoles 2024 Q1 Irrationality and Transcendence Proofs View
For a strictly positive rational number $a \in \mathbf{Q}_{>0}$, let $\log(a)$ denote the unique real number satisfying $e^{\log a} = a$. Deduce from Theorem 1 that $\log(a)$ is irrational for every strictly positive rational number $a \neq 1$.
(Theorem 1: Let $r \geq 2$ be an integer. If $a_1, \ldots, a_r \in \mathbf{Q}$ are distinct rational numbers, then the real numbers $e^{a_1}, \ldots, e^{a_r}$ are linearly independent over $\mathbf{Q}$.)
grandes-ecoles 2024 Q6 Algebraic Number Theory and Minimal Polynomials View
Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients whose constant term equals 1. Show that there exists an integer $b \geq 1$ such that $Q(bx)$ has integer coefficients.
grandes-ecoles 2024 Q16 Divisibility and Divisor Analysis View
Show that $h(x) = \sum_{n=0}^{\infty} \frac{(2n)!(3n)!}{(n!)^5} x^n$ has integer coefficients.
grandes-ecoles 2024 Q17a 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 $n \in \{1, 2, 3\}$.
grandes-ecoles 2024 Q17b 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 now 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.
grandes-ecoles 2024 Q17c 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 now 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 $\prod_{\substack{m+1 < p \leqslant 2m+1 \\ p \text{ prime}}} p$ divides $\binom{2m+1}{m}$ and show that $\binom{2m+1}{m} \leqslant 4^{m}$.
grandes-ecoles 2024 Q17d 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.
grandes-ecoles 2024 Q17a 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 $n \in \{1, 2, 3\}$.
grandes-ecoles 2024 Q17b 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.
grandes-ecoles 2024 Q17c 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 $\binom{2m+1}{m} \leqslant 4^m$.
grandes-ecoles 2024 Q17d 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.
grandes-ecoles 2024 Q18 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 $$\frac{n}{p} - 1 < \nu_{p}(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$
grandes-ecoles 2024 Q18 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 $$\frac{n}{p} - 1 < \nu_p(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$
grandes-ecoles 2024 Q19b Prime Counting and Distribution View
Justify that $n! = \prod_{\substack{p \leqslant n \\ p \text{ prime}}} p^{\nu_{p}(n!)}$ and deduce that $$n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} - n\ln(4) < \ln(n!) \leqslant n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} + n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p(p-1)}.$$
grandes-ecoles 2024 Q19d Prime Counting and Distribution View
Conclude that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} \underset{n \rightarrow +\infty}{=} \ln(n) + O(1)$.
grandes-ecoles 2024 Q19b Prime Counting and Distribution View
Justify that $n! = \prod_{\substack{p \leqslant n \\ p \text{ prime}}} p^{\nu_p(n!)}$ and deduce that $$n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} - n\ln(4) < \ln(n!) \leqslant n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} + n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p(p-1)}.$$
grandes-ecoles 2024 Q19d Prime Counting and Distribution View
Conclude that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} \underset{n \rightarrow +\infty}{=} \ln(n) + O(1)$.
grandes-ecoles 2024 Q20a Prime Counting and Distribution View
For all real $t \geqslant 2$, we define $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Show, using the result from question 16, that $$\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} = 1 + \ln_{2}(n) - \ln_{2}(2) + \frac{R(n)}{\ln(n)} + \int_{2}^{n} \frac{R(t)}{t(\ln(t))^{2}} \mathrm{~d}t$$

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