# Number Theory

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Paper 1, Section I, 1I

commentState Euler's criterion.

Let $p$ be an odd prime. Show that every primitive root modulo $p$ is a quadratic non-residue modulo $p$.

Let $p$ be a Fermat prime, that is, a prime of the form $2^{2^{k}}+1$ for some $k \geqslant 1$. By evaluating $\phi(p-1)$, or otherwise, show that every quadratic non-residue modulo $p$ is a primitive root modulo $p$. Deduce that 3 is a primitive root modulo $p$ for every Fermat prime $p$.

Paper 2, Section $I$, I

commentDefine the Möbius function $\mu$, and explain what it means for it to be multiplicative.

Show that for every positive integer $n$

$\sum_{d \mid n} \frac{\mu(d)^{2}}{\phi(d)}=\frac{n}{\phi(n)}$

where $\phi$ is the Euler totient function.

Fix an integer $k \geqslant 1$. Use the Chinese remainder theorem to show that there are infinitely many positive integers $n$ for which

$\mu(n)=\mu(n+1)=\cdots=\mu(n+k)$

Paper 3, Section I, I

commentDefine the continued fraction expansion of $\theta \in \mathbb{R}$, and show that this expansion terminates if and only if $\theta \in \mathbb{Q}$.

Define the convergents $\left(p_{n} / q_{n}\right)_{n \geqslant-1}$ of the continued fraction expansion of $\theta$, and show that for all $n \geqslant 0$,

$p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n-1} .$

Deduce that if $\theta \in \mathbb{R} \backslash \mathbb{Q}$, then for all $n \geqslant 0$, at least one of

$\left|\theta-\frac{p_{n}}{q_{n}}\right|<\frac{1}{2 q_{n}^{2}} \quad \text { and } \quad\left|\theta-\frac{p_{n+1}}{q_{n+1}}\right|<\frac{1}{2 q_{n+1}^{2}}$

must hold.

[You may assume that $\theta$ lies strictly between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1}$ for all $n \geqslant 0 .$ ]

Paper 3, Section II, I

commentState what it means for two binary quadratic forms to be equivalent, and define the class number $h(d)$.

Let $m$ be a positive integer, and let $f$ be a binary quadratic form. Show that $f$ properly represents $m$ if and only if $f$ is equivalent to a binary quadratic form

$m x^{2}+b x y+c y^{2}$

for some integers $b$ and $c$.

Let $d<0$ be an integer such that $d \equiv 0$ or $1 \bmod 4$. Show that $m$ is properly represented by some binary quadratic form of discriminant $d$ if and only if $d$ is a square modulo $4 m$.

Fix a positive integer $A \geqslant 2$. Show that $n^{2}+n+A$ is composite for some integer $n$ such that $0 \leqslant n \leqslant A-2$ if and only if $d=1-4 A$ is a square modulo $4 p$ for some prime $p<A$.

Deduce that $h(1-4 A)=1$ if and only if $n^{2}+n+A$ is prime for all $n=0,1, \ldots, A-2$.

Paper 4, Section I, I

commentLet $p$ be a prime, and let $N=\left(\begin{array}{c}2 n \\ n\end{array}\right)$ for some positive integer $n$.

Show that if a prime power $p^{k}$ divides $N$ for some $k \geqslant 1$, then $p^{k} \leqslant 2 n$.

Given a positive real $x$, define $\psi(x)=\sum_{n \leqslant x} \Lambda(n)$, where $\Lambda(n)$ is the von Mangoldt function, taking the value $\log p$ if $n=p^{k}$ for some prime $p$ and integer $k \geqslant 1$, and 0 otherwise. Show that

$\psi(x)=\sum_{p \leqslant x, p \text { prime }}\left\lfloor\frac{\log x}{\log p}\right\rfloor \log p$

Deduce that for all integers $n>1, \psi(2 n) \geqslant n \log 2$.

Paper 4, Section II, I

comment(a) Let $N \geqslant 3$ be an odd integer and $b$ an integer with $(b, N)=1$. What does it mean to say that $N$ is a (Fermat) pseudoprime to base b?

Let $b, k \geqslant 2$ be integers. Show that if $N \geqslant 3$ is an odd composite integer dividing $b^{k}-1$ and satisfying $N \equiv 1 \bmod k$, then $N$ is a pseudoprime to base $b$.

(b) Fix $b \geqslant 2$. Let $p$ be an odd prime not dividing $b^{2}-1$, and let

$n=\frac{b^{p}-1}{b-1} \quad \text { and } \quad m=\frac{b^{p}+1}{b+1} .$

Use the conclusion of part (a) to show that $N=n m$ is a pseudoprime to base $b$. Deduce that there are infinitely many pseudoprimes to base $b$.

(c) Let $b, k \geqslant 2$ be integers, and let $n=p_{1} \cdots p_{k}$, where $p_{1}, p_{2}, \ldots, p_{k}$ are distinct primes not dividing $2 b$. For each $j=1,2, \ldots, k$, let $r_{j}=n / p_{j}$. Show that $n$ is a pseudoprime to base $b$ if and only if for all $j=1,2, \ldots, k$, the order of $b$ modulo $p_{j}$ divides $r_{j}-1$.

(d) By considering products of prime factors of $2^{k}-1$ and $2^{k}+1$ for primes $k \geqslant 5$, deduce that there are infinitely many pseudoprimes to base 2 with two prime factors.

[Hint: You may assume that $\operatorname{gcd}(j, k)=1$ for $j, k \geqslant 1$ implies $\operatorname{gcd}\left(2^{j}-1,2^{k}-1\right)=1$, and that for $k>3,2^{k}+1$ is not a power of 3.]

Paper 1, Section I, H

commentWhat does it mean to say that a positive definite binary quadratic form is reduced?

Find all reduced binary quadratic forms of discriminant $-20$.

Prove that if a prime $p \neq 5$ is represented by $x^{2}+5 y^{2}$, then $p \equiv 1,3,7$ or $9 \bmod 20$.

Paper 2, Section I, $1 \mathbf{H}$

commentLet $\theta \in \mathbb{R}$.

For each integer $n \geqslant-1$, define the convergents $p_{n} / q_{n}$ of the continued fraction expansion of $\theta$. Show that for all $n \geqslant 0, p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n-1}$. Deduce that if $q \in \mathbb{N}$ and $p \in \mathbb{Z}$ satisfy

$\left|\theta-\frac{p}{q}\right|<\left|\theta-\frac{p_{n}}{q_{n}}\right|$

then $q>q_{n}$.

Compute the continued fraction expansion of $\sqrt{12}$. Hence or otherwise find a solution in positive integers $x$ and $y$ to the equation $x^{2}-12 y^{2}=1$.

Paper 3, Section I, $1 \mathrm{H}$

commentLet $N \geqslant 3$ be an odd integer and $b$ an integer with $(b, N)=1$. What does it mean to say that $N$ is an Euler pseudoprime to base $b$ ?

Show that if $N$ is not an Euler pseudoprime to some base $b_{0}$, then it is not an Euler pseudoprime to at least half the bases $\{1 \leqslant b<N:(b, N)=1\}$.

Show that if $N$ is odd and composite, then there exists an integer $b$ such that $N$ is not an Euler pseudoprime to base $b$.

Paper 3, Section II, 11H

commentLet $p$ be an odd prime.

(i) Define the Legendre symbol $\left(\frac{x}{p}\right)$, and show that when $(x, p)=1$, then $\left(\frac{x^{-1}}{p}\right)=\left(\frac{x}{p}\right)$.

(ii) State and prove Gauss's lemma, and use it to evaluate $\left(\frac{-1}{p}\right)$. [You may assume Euler's criterion.]

(iii) Prove that

$\sum_{x=1}^{p}\left(\frac{x}{p}\right)=0$

and deduce that

$\sum_{x=1}^{p}\left(\frac{x(x+1)}{p}\right)=-1$

Hence or otherwise determine the number of pairs of consecutive integers $z, z+1$ such that $1 \leqslant z, z+1 \leqslant p-1$ and both $z$ and $z+1$ are quadratic residues $\bmod p$.

Paper 4 , Section II, 11H

comment(a) What does it mean to say that a function $f: \mathbb{N} \rightarrow \mathbb{C}$ is multiplicative? Show that if $f, g: \mathbb{N} \rightarrow \mathbb{C}$ are both multiplicative, then so is $f \star g: \mathbb{N} \rightarrow \mathbb{C}$, defined for all $n \in \mathbb{N}$ by

$f \star g(n)=\sum_{d \mid n} f(d) g\left(\frac{n}{d}\right)$

Show that if $f=\mu \star g$, where $\mu$ is the Möbius function, then $g=f \star 1$, where 1 denotes the constant function 1 .

(b) Let $\tau(n)$ denote the number of positive divisors of $n$. Find $f, g: \mathbb{N} \rightarrow \mathbb{C}$ such that $\tau=f \star g$, and deduce that $\tau$ is multiplicative. Hence or otherwise show that for all $s \in \mathbb{C}$ with $\operatorname{Re}(s)>1$,

$\sum_{n=1}^{\infty} \frac{\tau(n)}{n^{s}}=\zeta(s)^{2}$

where $\zeta$ is the Riemann zeta function.

(c) Fix $k \in \mathbb{N}$. By considering suitable powers of the product of the first $k+1$ primes, show that

$\tau(n) \geqslant(\log n)^{k}$

for infinitely many $n \in \mathbb{N}$.

(d) Fix $\epsilon>0$. Show that

$\frac{\tau(n)}{n^{\epsilon}}=\prod_{p \text { prime, } p^{\alpha}|| n} \frac{(\alpha+1)}{p^{\alpha \epsilon}}$

where $p^{\alpha} \| n$ denotes the fact that $\alpha \in \mathbb{N}$ is such that $p^{\alpha} \mid n$ but $p^{\alpha+1} \nmid n$. Deduce that there exists a positive constant $C(\epsilon)$ depending only on $\epsilon$ such that for all $n \in \mathbb{N}, \tau(n) \leqslant C(\epsilon) n^{\epsilon} .$

Paper 4, Section I, H

commentLet $p$ be a prime.

State and prove Lagrange's theorem on the number of solutions of a polynomial congruence modulo $p$. Deduce that $(p-1) ! \equiv-1 \bmod p$.

Let $k$ be a positive integer such that $k \mid(p-1)$. Show that the congruence

$x^{k} \equiv 1 \quad \bmod p$

has precisely $k$ solutions modulo $p$.

Paper 1, Section I, I

comment(a) State and prove the Chinese remainder theorem.

(b) Let $N$ be an odd positive composite integer, and $b$ a positive integer with $(b, N)=1$. What does it mean to say that $N$ is a Fermat pseudoprime to base b? Show that 35 is a Fermat pseudoprime to base $b$ if and only if $b$ is congruent to one of $1,6,29$ or $34(\bmod 35)$.

Paper 2, Section I, I

commentDefine the Jacobi symbol $\left(\frac{a}{n}\right)$, where $a, n \in \mathbb{Z}$ and $n$ is odd and positive.

State and prove the Law of Quadratic Reciprocity for the Jacobi symbol. [You may use Quadratic Reciprocity for the Legendre symbol without proof but should state it clearly.]

Compute the Jacobi symbol $\left(\frac{503}{2019}\right)$.

Paper 3, Section I, I

commentLet $f=(a, b, c)$ be a positive definite binary quadratic form with integer coefficients. What does it mean to say that $f$ is reduced? Show that if $f$ is reduced and has discriminant $d$, then $|b| \leqslant a \leqslant \sqrt{|d| / 3}$ and $b \equiv d(\bmod 2)$. Deduce that for fixed $d<0$, there are only finitely many reduced $f$ of discriminant $d$.

Find all reduced positive definite binary quadratic forms of discriminant $-15$.

Paper 3, Section II, I

commentLet $p>2$ be a prime.

(a) What does it mean to say that an integer $g$ is a primitive root $\bmod p$ ?

(b) Let $k$ be an integer with $0 \leqslant k<p-1$. Let

$S_{k}=\sum_{x=0}^{p-1} x^{k}$

Show that $S_{k} \equiv 0(\bmod p)$. [Recall that by convention $0^{0}=1$.]

(c) Let $f(X, Y, Z)=a X^{2}+b Y^{2}+c Z^{2}$ for some $a, b, c \in \mathbb{Z}$, and let $g=1-f^{p-1}$. Show that for any $x, y, z \in \mathbb{Z}, g(x, y, z) \equiv 0$ or $1(\bmod p)$, and that

$\sum_{x, y, z \in\{0,1, \ldots, p-1\}} g(x, y, z) \equiv 0 \quad(\bmod p) .$

Hence show that there exist integers $x, y, z$, not all divisible by $p$, such that $f(x, y, z) \equiv 0$ $(\bmod p)$.

Paper 4, Section I, I

commentShow that the product

$\prod_{p \text { prime }}\left(1-\frac{1}{p}\right)^{-1}$

and the series

$\sum_{p \text { prime }} \frac{1}{p}$

are both divergent.

Paper 4, Section II, I

comment(a) Let $a_{0}, a_{1}, \ldots$ be positive integers, and $\beta>0$ a positive real number. Show that for every $n \geqslant 0$, if $\theta_{n}=\left[a_{0}, \ldots, a_{n}, \beta\right]$, then $\theta_{n}=\left(\beta p_{n}+p_{n-1}\right) /\left(\beta q_{n}+q_{n-1}\right)$, where $\left(p_{n}\right)$, $\left(q_{n}\right)(n \geqslant-1)$ are sequences of integers satisfying

$\begin{aligned} &p_{0}=a_{0}, q_{0}=1, \quad p_{-1}=1, \quad q_{-1}=0 \quad \text { and } \\ &\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} p_{n-1} & p_{n-2} \\ q_{n-1} & q_{n-2} \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right) \quad(n \geqslant 1) \end{aligned}$

Show that $p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n-1}$, and that $\theta_{n}$ lies between $p_{n} / q_{n}$ and $p_{n-1} / q_{n-1}$.

(b) Show that if $\left[a_{0}, a_{1}, \ldots\right]$ is the continued fraction expansion of a positive irrational $\theta$, then $p_{n} / q_{n} \rightarrow \theta$ as $n \rightarrow \infty$.

(c) Let the convergents of the continued fraction $\left[a_{0}, a_{1}, \ldots, a_{n}\right]$ be $p_{j} / q_{j}(0 \leqslant$ $j \leqslant n)$. Using part (a) or otherwise, show that the $n$-th and $(n-1)$-th convergents of $\left[a_{n}, a_{n-1}, \ldots, a_{0}\right]$ are $p_{n} / p_{n-1}$ and $q_{n} / q_{n-1}$ respectively.

(d) Show that if $\theta=\left[\overline{a_{0}, a_{1}, \ldots, a_{n}}\right]$ is a purely periodic continued fraction with convergents $p_{j} / q_{j}$, then $f(\theta)=0$, where $f(X)=q_{n} X^{2}+\left(q_{n-1}-p_{n}\right) X-p_{n-1}$. Deduce that if $\theta^{\prime}$ is the other root of $f(X)$, then $-1 / \theta^{\prime}=\left[\overline{a_{n}, a_{n-1}, \ldots, a_{0}}\right]$.

Paper 1, Section I, G

comment(a) State and prove the Chinese remainder theorem.

(b) An integer $n$ is squarefull if whenever $p$ is prime and $p \mid n$, then $p^{2} \mid n$. Show that there exist 1000 consecutive positive integers, none of which are squarefull.

Paper 2, Section I, G

commentDefine the Legendre symbol, and state Gauss's lemma. Show that if $p$ is an odd prime, then

$\left(\frac{2}{p}\right)=(-1)^{\left(p^{2}-1\right) / 8}$

Use the law of quadratic reciprocity to compute $\left(\frac{105}{149}\right)$.

Paper 3, Section I, G

commentWhat is a multiplicative function? Show that if $f(n)$ is a multiplicative function, then so is $g(n)=\sum_{d \mid n} f(d)$.

Define the Möbius function $\mu(n)$, and show that it is multiplicative. Deduce that

$\sum_{d \mid n} \mu(d)= \begin{cases}1 & \text { if } n=1 \\ 0 & \text { if } n>1\end{cases}$

and that

$f(n)=\sum_{e \mid n} \mu(e) g\left(\frac{n}{e}\right) .$

What is $g(n)$ if $f(n)=n ?$ What is $f(n)$ if $g(n)=n ?$

Paper 3, Section II, G

commentWhat does it mean to say that a positive definite binary quadratic form is reduced? What does it mean to say that two binary quadratic forms are equivalent? Show that every positive definite binary quadratic form is equivalent to some reduced form.

Show that the reduced positive definite binary quadratic forms of discriminant $-35$ are $f_{1}=x^{2}+x y+9 y^{2}$ and $f_{2}=3 x^{2}+x y+3 y^{2}$. Show also that a prime $p>7$ is represented by $f_{i}$ if and only if

$\left(\frac{p}{5}\right)=\left(\frac{p}{7}\right)= \begin{cases}+1 & (i=1) \\ -1 & (i=2)\end{cases}$

Paper 4, Section I, G

commentShow that if a continued fraction is periodic, then it represents a quadratic irrational. What number is represented by the continued fraction $[7,7,7, \ldots]$ ?

Compute the continued fraction expansion of $\sqrt{23}$. Hence or otherwise find a solution in positive integers to the equation $x^{2}-23 y^{2}=1$.

Paper 4, Section II, G

comment(a) State and prove the Fermat-Euler theorem. Let $p$ be a prime and $k$ a positive integer. Show that $b^{k} \equiv b(\bmod p)$ holds for every integer $b$ if and only if $k \equiv 1(\bmod p-1)$.

(b) Let $N \geqslant 3$ be an odd integer and $b$ be an integer with $(b, N)=1$. What does it mean to say that $N$ is a Fermat pseudoprime to base $b$ ? What does it mean to say that $N$ is a Carmichael number?

Show that every Carmichael number is squarefree, and that if $N$ is squarefree, then $N$ is a Carmichael number if and only if $N \equiv 1(\bmod p-1)$ for every prime divisor $p$ of $N$. Deduce that a Carmichael number is a product of at least three primes.

(c) Let $r$ be a fixed odd prime. Show that there are only finitely many pairs of primes $p, q$ for which $N=p q r$ is a Carmichael number.

[You may assume throughout that $\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)^{*}$ is cyclic for every odd prime $p$ and every integer $n \geqslant 1 .]$

Paper 1, Section $I$, G

commentDefine the Legendre symbol $\left(\frac{a}{p}\right)$.

State Gauss' lemma and use it to compute $\left(\frac{2}{p}\right)$ where $p$ is an odd prime.

Show that if $m \geqslant 4$ is a power of 2 , and $p$ is a prime dividing $2^{m}+1$, then $p \equiv 1(\bmod 4 m)$.

Paper 2, Section I, G

commentState and prove Legendre's formula for $\pi(x)$. Use it to compute $\pi(42)$.

Paper 3, Section I, G

commentExplain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler pseudoprime to the base $b$ if and only if $b^{2} \equiv \pm 1(\bmod 65)$. How many such bases are there? Show that the bases for which 65 is a strong pseudoprime do not form a subgroup of $(\mathbb{Z} / 65 \mathbb{Z})^{\times}$.

Paper 3, Section II, G

commentLet $d$ be a positive integer which is not a square. Assume that the continued fraction expansion of $\sqrt{d}$ takes the form $\left[a_{0}, \overline{a_{1}, a_{2}, \ldots, a_{m}}\right]$.

(a) Define the convergents $p_{n} / q_{n}$, and show that $p_{n}$ and $q_{n}$ are coprime.

(b) The complete quotients $\theta_{n}$ may be written in the form $\left(\sqrt{d}+r_{n}\right) / s_{n}$, where $r_{n}$ and $s_{n}$ are rational numbers. Use the relation

$\sqrt{d}=\frac{\theta_{n} p_{n-1}+p_{n-2}}{\theta_{n} q_{n-1}+q_{n-2}}$

to find formulae for $r_{n}$ and $s_{n}$ in terms of the $p$ 's and $q$ 's. Deduce that $r_{n}$ and $s_{n}$ are integers.

(c) Prove that Pell's equation $x^{2}-d y^{2}=1$ has infinitely many solutions in integers $x$ and $y$.

(d) Find integers $x$ and $y$ satisfying $x^{2}-67 y^{2}=-2$.

Paper 4, Section I, G

commentShow that, for $x \geqslant 2$ a real number,

$\prod_{\substack{p \leqslant x \\ p \text { is prime }}}\left(1-\frac{1}{p}\right)^{-1}>\log x$

Hence prove that

$\sum_{\substack{p \leqslant x, p \text { is prime }}} \frac{1}{p}>\log \log x+c,$

where $c$ is a constant you should make explicit.

Paper 4, Section II, 10G

comment(a) State Dirichlet's theorem on primes in arithmetic progression.

(b) Let $d$ be the discriminant of a binary quadratic form, and let $p$ be an odd prime. Show that $p$ is represented by some binary quadratic form of discriminant $d$ if and only if $x^{2} \equiv d(\bmod p)$ is soluble.

(c) Let $f(x, y)=x^{2}+15 y^{2}$ and $g(x, y)=3 x^{2}+5 y^{2}$. Show that $f$ and $g$ each represent infinitely many primes. Are there any primes represented by both $f$ and $g$ ?

Paper 1, Section I, I

commentDefine the Riemann zeta function $\zeta(s)$ for $\operatorname{Re}(s)>1$. State and prove the alternative formula for $\zeta(s)$ as an Euler product. Hence or otherwise show that $\zeta(s) \neq 0$ for $\operatorname{Re}(s)>1$.

Paper 2, Section I, I

commentDefine the Legendre symbol and the Jacobi symbol. Compute the Jacobi symbols $\left(\frac{202}{11189}\right)$ and $\left(\frac{974}{1001}\right)$, stating clearly any properties of these symbols that you use.

Paper 3, Section I, I

commentShow that the exact power of a prime $p$ dividing $N !$ is $\sum_{j=1}^{\infty}\left\lfloor\frac{N}{p^{j}}\right\rfloor$. By considering the prime factorisation of $\left(\begin{array}{c}2 n \\ n\end{array}\right)$, show that

$\frac{4^{n}}{2 n+1} \leqslant\left(\begin{array}{c} 2 n \\ n \end{array}\right) \leqslant(2 n)^{\pi(2 n)}$

Setting $n=\left\lfloor\frac{x}{2}\right\rfloor$, deduce that for $x$ sufficiently large

$\pi(x)>\frac{\left\lfloor\frac{x}{2}\right\rfloor \log 3}{\log x}>\frac{x}{2 \log x}$

Paper 3, Section II, I

commentWhat does it mean for a positive definite binary quadratic form to be reduced?

Prove that every positive definite binary quadratic form is equivalent to a reduced form, and that there are only finitely many reduced forms with given discriminant.

State a criterion for a positive integer $n$ to be represented by a positive definite binary quadratic form with discriminant $d<0$, and hence determine which primes $p$ are represented by $x^{2}+x y+7 y^{2}$.

Paper 4 , Section I, I

commentCompute the continued fraction expansion of $\sqrt{14}$, and use it to find two solutions to $x^{2}-14 y^{2}=2$ where $x$ and $y$ are positive integers.

Paper 4, Section II, 10I

comment(a) Define Euler's totient function $\phi(n)$ and show that $\sum_{d \mid n} \phi(d)=n$.

(b) State Lagrange's theorem concerning roots of polynomials mod $p$.

(c) Let $p$ be a prime. Proving any results you need about primitive roots, show that $x^{m} \equiv 1(\bmod p)$ has exactly $(m, p-1)$ roots.

(d) Show that if $p$ and $3 p-2$ are both primes then $N=p(3 p-2)$ is a Fermat pseudoprime for precisely a third of all bases.

Paper 1, Section I, H

commentDefine the Legendre symbol $\left(\frac{a}{p}\right)$. State and prove Euler's criterion, assuming if you wish the existence of primitive roots $\bmod p$.

By considering the prime factors of $n^{2}+4$ for $n$ an odd integer, prove that there are infinitely many primes $p$ with $p \equiv 5(\bmod 8)$.

Paper 2, Section I, H

commentDefine the Euler totient function $\phi$ and the Möbius function $\mu$. Suppose $f$ and $g$ are functions defined on the natural numbers satisfying $f(n)=\sum_{d \mid n} g(d)$. State and prove a formula for $g$ in terms of $f$. Find a relationship between $\mu$ and $\phi$.

Define the Riemann zeta function $\zeta(s)$. Find a Dirichlet series for $\zeta(s-1) / \zeta(s)$ valid for $\operatorname{Re}(s)>2$.

Paper 3, Section I, H

commentWhat does it mean to say that a positive definite binary quadratic form is reduced? Find the three smallest positive integers properly represented by each of the forms $f(x, y)=3 x^{2}+8 x y+9 y^{2}$ and $g(x, y)=15 x^{2}+34 x y+20 y^{2}$. Show that every odd integer represented by some positive definite binary quadratic form with discriminant $-44$ is represented by at least one of the forms $f$ and $g$.

Paper 3, Section II, H

commentLet $\theta$ be a real number with continued fraction expansion $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$. Define the convergents $p_{n} / q_{n}$ (by means of recurrence relations) and show that for $\beta>0$ we have

$\left[a_{0}, a_{1}, \ldots, a_{n-1}, \beta\right]=\frac{\beta p_{n-1}+p_{n-2}}{\beta q_{n-1}+q_{n-2}}$

Show that

$\left|\theta-\frac{p_{n}}{q_{n}}\right|<\frac{1}{q_{n} q_{n+1}}$

and deduce that $p_{n} / q_{n} \rightarrow \theta$ as $n \rightarrow \infty$.

By computing a suitable continued fraction expansion, find solutions in positive integers $x$ and $y$ to each of the equations $x^{2}-53 y^{2}=4$ and $x^{2}-53 y^{2}=-7$.

Paper 4, Section I, H

commentShow that if $10^{n}+1$ is prime then $n$ must be a power of 2 . Now assuming $n$ is a power of 2 , show that if $p$ is a prime factor of $10^{n}+1$ then $p \equiv 1(\bmod 2 n)$.

Explain the method of Fermat factorization, and use it to factor $10^{4}+1$.

Paper 4, Section II, H

commentState the Chinese Remainder Theorem.

Let $N$ be an odd positive integer. Define the Jacobi symbol $\left(\frac{a}{N}\right)$. Which of the following statements are true, and which are false? Give a proof or counterexample as appropriate.

(i) If $\left(\frac{a}{N}\right)=1$ then the congruence $x^{2} \equiv a(\bmod N)$ is soluble.

(ii) If $N$ is not a square then $\sum_{a=1}^{N}\left(\frac{a}{N}\right)=0$.

(iii) If $N$ is composite then there exists an integer a coprime to $N$ with

$a^{N-1} \not \equiv 1 \quad(\bmod N)$

(iv) If $N$ is composite then there exists an integer $a$ coprime to $N$ with

$a^{(N-1) / 2} \not \equiv\left(\frac{a}{N}\right) \quad(\bmod N)$

Paper 1, Section I, $1 F$

commentDefine what it means for a number $N$ to be a pseudoprime to the base $b$.

Show that if there is a base $b$ to which $N$ is not a pseudoprime, then $N$ is a pseudoprime to at most half of all possible bases.

Let $n$ be an integer greater than 1 such that $F_{n}=2^{2^{n}}+1$ is composite. Show that $F_{n}$ is a pseudoprime to the base 2 .

Paper 2, Section I, F

commentShow that

$\sum_{p \leqslant x} \frac{1}{p} \geqslant \log \log x-\frac{1}{2}$

Deduce that there are infinitely many primes.

Paper 3, Section I, $1 F$

commentShow that the continued fraction for $\sqrt{51}$ is $[7 ; \overline{7,14}]$.

Hence, or otherwise, find positive integers $x$ and $y$ that satisfy the equation $x^{2}-51 y^{2}=1 .$

Are there integers $x$ and $y$ such that $x^{2}-51 y^{2}=-1$ ?

Paper 3, Section II, F

commentState and prove Lagrange's theorem about polynomial congruences modulo a prime.

Define the Euler totient function $\phi$.

Let $p$ be a prime and let $d$ be a positive divisor of $p-1$. Show that there are exactly $\phi(d)$ elements of $(\mathbb{Z} / p \mathbb{Z})^{\times}$with order $d .$

Deduce that $(\mathbb{Z} / p \mathbb{Z})^{\times}$is cyclic.

Let $g$ be a primitive root modulo $p^{2}$. Show that $g$ must be a primitive root modulo $p$.

Let $g$ be a primitive root modulo $p$. Must it be a primitive root modulo $p^{2}$ ? Give a proof or a counterexample.

Paper 4, Section I, F

commentState the Chinese Remainder Theorem.

Find all solutions to the simultaneous congruences

$\begin{aligned} &x \equiv 2 \quad(\bmod 3) \\ &x \equiv 3(\bmod 5) \\ &x \equiv 5(\bmod 7) \end{aligned}$

A positive integer is said to be square-free if it is the product of distinct primes. Show that there are 100 consecutive numbers that are not square-free.

Paper 4, Section II, F

commentDefine the Legendre and Jacobi symbols.

State the law of quadratic reciprocity for the Legendre symbol.

State the law of quadratic reciprocity for the Jacobi symbol, and deduce it from the corresponding result for the Legendre symbol.

Let $p$ be a prime with $p \equiv 1(\bmod 4)$. Prove that the sum of the quadratic residues in the set $\{1,2, \ldots, p-1\}$ is equal to the sum of the quadratic non-residues in this set.

For which primes $p$ is 7 a quadratic residue?

Paper 1, Section I, I

commentState and prove Gauss's Lemma for the Legendre symbol $\left(\frac{a}{p}\right)$. For which odd primes $p$ is 2 a quadratic residue modulo $p ?$ Justify your answer.

Paper 2, Section I, I

commentDefine Euler's totient function $\phi(n)$, and show that $\sum_{d \mid n} \phi(d)=n$. Hence or otherwise prove that for any prime $p$ the multiplicative group $(\mathbb{Z} / p \mathbb{Z})^{\times}$is cyclic.

Paper 3, Section I, I

commentState the Chinese Remainder Theorem.

A composite number $n$ is defined to be a Carmichael number if $b^{n-1} \equiv 1 \bmod n$ whenever $(b, n)=1$. Show that a composite $n$ is Carmichael if and only if $n$ is square-free and $(p-1)$ divides $(n-1)$ for all prime factors $p$ of $n$. [You may assume that, for $p$ an odd prime and $\alpha \geqslant 1$ an integer, $\left(\mathbb{Z} / p^{\alpha} \mathbb{Z}\right)^{\times}$is a cyclic group.]

Show that if $n=(6 t+1)(12 t+1)(18 t+1)$ with all three factors prime, then $n$ is Carmichael.

Paper 3, Section II, I

commentDefine equivalence of binary quadratic forms and show that equivalent forms have the same discriminant.

Show that an integer $n$ is properly represented by a binary quadratic form of discriminant $d$ if and only if $x^{2} \equiv d \bmod 4 n$ is soluble in integers. Which primes are represented by a form of discriminant $-20$ ?

What does it mean for a positive definite form to be reduced? Find all reduced forms of discriminant $-20$. For each member of your list find the primes less than 100 represented by the form.

Paper 4, Section I, I

commentLet $s=\sigma+i t$ with $\sigma, t \in \mathbb{R}$. Define the Riemann zeta function $\zeta(s)$ for $\sigma>1$. Show that for $\sigma>1$,

$\zeta(s)=\prod_{p}\left(1-p^{-s}\right)^{-1}$

where the product is taken over all primes. Deduce that there are infinitely many primes.

Paper 4, Section II, I

comment(i) What is meant by the continued fraction expansion of a real number $\theta$ ? Suppose that $\theta$ has continued fraction $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$. Define the convergents $p_{n} / q_{n}$ to $\theta$ and give the recurrence relations satisfied by the $p_{n}$ and $q_{n}$. Show that the convergents $p_{n} / q_{n}$ do indeed converge to $\theta$.

[You need not justify the basic order properties of finite continued fractions.]

(ii) Find two solutions in strictly positive integers to each of the equations

$x^{2}-10 y^{2}=1 \quad \text { and } \quad x^{2}-11 y^{2}=1$

Paper 1, Section I, I

commentShow that the continued fraction for $\sqrt{13}$ is $[3 ; \overline{1,1,1,1,6}]$.

Hence, or otherwise, find a solution to the equation $x^{2}-13 y^{2}=1$ in positive integers $x$ and $y$. Write down an expression for another solution.

Paper 2, Section I, I

commentDefine the Legendre symbol and the Jacobi symbol.

State the law of quadratic reciprocity for the Jacobi symbol.

Compute the value of the Jacobi symbol $\left(\frac{247}{321}\right)$, stating clearly any results you use.

Paper 3 , Section II, I

commentLet $p$ be an odd prime. Prove that the multiplicative groups $\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)^{\times}$are cyclic for $n \geqslant 2$. [You may assume that the multiplicative group $(\mathbb{Z} / p \mathbb{Z})^{\times}$is cyclic.]

Find an integer which generates $\left(\mathbb{Z} / 7^{n} \mathbb{Z}\right)^{\times}$for all $n \geqslant 1$, justifying your answer.

Paper 3, Section I, I

commentDefine the discriminant of the binary quadratic form $f(x, y)=a x^{2}+b x y+c y^{2}$.

Assuming that this form is positive definite, define what it means for $f$ to be reduced.

Show that there are precisely two reduced positive definite binary quadratic forms of discriminant $-35$.

Paper 4, Section I, I

commentDefine what it means for the composite natural number $N$ to be a pseudoprime to the base $b$.

Find the number of bases (less than 21) to which 21 is a pseudoprime. [You may, if you wish, assume the Chinese Remainder Theorem.]

Paper 4, Section II, I

commentLet $f: \mathbb{N} \rightarrow \mathbb{R}$ be a function, where $\mathbb{N}$ denotes the (positive) natural numbers.

Define what it means for $f$ to be a multiplicative function.

Prove that if $f$ is a multiplicative function, then the function $g: \mathbb{N} \rightarrow \mathbb{R}$ defined by

$g(n)=\sum_{d \mid n} f(d)$

is also multiplicative.

Define the Möbius function $\mu$. Is $\mu$ multiplicative? Briefly justify your answer.

Compute

$\sum_{d \mid n} \mu(d)$

for all positive integers $n$.

Define the Riemann zeta function $\zeta$ for complex numbers $s$ with $\Re(s)>1$.

Prove that if $s$ is a complex number with $\Re(s)>1$, then

$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}$

Paper 1, Section I, I

commentProve that, under the action of $\mathrm{SL}_{2}(\mathbb{Z})$, every positive definite binary quadratic form of discriminant $-163$, with integer coefficients, is equivalent to

$x^{2}+x y+41 y^{2} .$

Paper 2, Section I, I

comment(i) Find a primitive root modulo $17 .$

(ii) Let $p$ be a prime of the form $2^{m}+1$ for some integer $m \geqslant 1$. Prove that every quadratic non-residue modulo $p$ is a primitive root modulo $p$.

Paper 3, Section I, I

comment(i) State Lagrange's Theorem, and prove that, if $p$ is an odd prime,

$(p-1) ! \equiv-1 \quad \bmod p$

(ii) Still assuming $p$ is an odd prime, prove that

$3^{2} \cdot 5^{2} \cdots(p-2)^{2} \equiv(-1)^{\frac{p+1}{2}} \bmod p$

Paper 3, Section II, I

commentLet $\zeta(s)$ be the Riemann zeta function, and put $s=\sigma+i t$ with $\sigma, t \in \mathbb{R}$.

(i) If $\sigma>1$, prove that

$\zeta(s)=\prod_{p}\left(1-p^{-s}\right)^{-1}$

where the product is taken over all primes $p$.

(ii) Assuming that, for $\sigma>1$, we have

$\zeta(s)=\sum_{n=1}^{\infty} n\left(n^{-s}-(n+1)^{-s}\right)$

prove that $\zeta(s)-\frac{1}{s-1}$ has an analytic continuation to the half plane $\sigma>0$.

Paper 4, Section $I$, I

comment(i) Prove that there are infinitely many primes.

(ii) Prove that arbitrarily large gaps can occur between consecutive primes.

Paper 4, Section II, I

comment(i) Prove the law of reciprocity for the Jacobi symbol. You may assume the law of reciprocity for the Legendre symbol.

(ii) Let $n$ be an odd positive integer which is not a square. Prove that there exists an odd prime $p$ with $\left(\frac{n}{p}\right)=-1$.

Paper 1, Section I, G

comment(i) Let $N$ be an integer $\geqslant 2$. Define the addition and multiplication on the set of congruence classes modulo $N$.

(ii) Let an integer $M \geqslant 1$ have expansion to the base 10 given by $a_{s} \ldots a_{0}$. Prove that 11 divides $M$ if and only if $\sum_{i=0}^{s}(-1)^{i} a_{i}$ is divisible by 11 .

Paper 2, Section I, G

commentLet $p$ be an odd prime number. If $n$ is an integer prime to $p$, define $\left(\frac{n}{p}\right)$.

(i) Prove that $\chi(n)=\left(\frac{n}{p}\right)$ defines a homomorphism from $(\mathbb{Z} / p \mathbb{Z})^{\times}$to the group $\{\pm 1\}$. What is the value of $\chi(-1) ?$

(ii) If $p \equiv 1 \bmod 4$, prove that $\sum_{n=1}^{p-1} \chi(n) n=0$.

Paper 3, Section I, G

comment(i) Let $M$ and $N$ be positive integers, such that $N$ is not a perfect square. If $M<\sqrt{N}$, show that every solution of the equation

$x^{2}-N y^{2}=M$

in positive integers $x, y$ comes from some convergent of the continued fraction of $\sqrt{N}$.

(ii) Find a solution in positive integers $x, y$ of

$x^{2}-29 y^{2}=5$

Paper 3, Section II, G

commentState precisely the Miller-Rabin primality test.

(i) Let $p$ be a prime $\geqslant 5$, and define

$N=\frac{4^{p}-1}{3}$

Prove that $N$ is a composite odd integer, and that $N$ is a pseudo-prime to the base 2 .

(ii) Let $M$ be an odd integer greater than 1 such that $M$ is a pseudo-prime to the base 2 . Prove that $2^{M}-1$ is always a strong pseudo-prime to the base 2 .

Paper 4, Section I, G

commentLet $p$ be a prime number, and put

$a_{k}=k p, \quad N_{k}=a_{k}^{p}-1 \quad(k=1,2, \ldots)$

Prove that $a_{k}$ has exact order $p$ modulo $N_{k}$ for all $k \geqslant 1$, and deduce that $N_{k}$ must be divisible by a prime $q$ with $q \equiv 1 \quad(\bmod p)$. By making a suitable choice of $k$, prove that there are infinitely many primes $q$ with $q \equiv 1 \quad(\bmod p)$.

Paper 4, Section II, G

commentLet $\mathcal{S}$ be the set of all positive definite binary quadratic forms with integer coefficients. Define the action of the group $S L_{2}(\mathbb{Z})$ on $\mathcal{S}$, and prove that equivalent forms under this action have the same discriminant.

Find necessary and sufficient conditions for an odd positive integer $n$, prime to 35 , to be properly represented by at least one of the two forms

$x^{2}+x y+9 y^{2}, \quad 3 x^{2}+x y+3 y^{2}$

Paper 1, Section I, G

commentState the Chinese Remainder Theorem.

Determine all integers $x$ satisfying the congruences $x \equiv 2 \bmod 3, x \equiv 2 \bmod 5$, $x \equiv 6 \bmod 7 .$

Paper 2, Section $\mathbf{I}$, $1 \mathrm{G}$

commentState the law of quadratic reciprocity for the Jacobi symbol $\left(\frac{m}{n}\right)$, where $m, n$ are odd positive integers, and prove this law using the reciprocity law for the Legendre symbol.

Compute the Jacobi symbol $\left(\frac{261}{317}\right)$.

Paper 3, Section I, G

commentFor any integer $x \geqslant 2$, define $\theta(x)=\sum_{p \leqslant x} \log p$, where the sum is taken over all primes $p \leqslant x$. Put $\theta(1)=0$. By studying the integer

$\left(\begin{array}{c} 2 n \\ n \end{array}\right)$

where $n \geqslant 1$ is an integer, prove that

$\theta(2 n)-\theta(n)<2 n \log 2$

Deduce that

$\theta(x)<(4 \log 2) x$

for all $x \geqslant 1$.

Paper 3, Section II, G

commentLet $p$ be an odd prime. Prove that there is an equal number of quadratic residues and non-residues in the set $\{1, \ldots, p-1\}$.

If $n$ is an integer prime to $p$, let $m_{n}$ be an integer such that $n m_{n} \equiv 1 \bmod p$. Prove that

$n(n+1) \equiv n^{2}\left(1+m_{n}\right) \bmod p$

and deduce that

$\sum_{n=1}^{p-1}\left(\frac{n(n+1)}{p}\right)=-1$

Paper 4, Section I, G

commentLet $W$ denote the set of all positive definite binary quadratic forms, with integer coefficients, and having discriminant $-67$. Let $S L_{2}(\mathbb{Z})$ be the group of all $2 \times 2$ matrices with integer entries and determinant 1. Prove that $W$ is infinite, but that all elements of $W$ are equivalent under the action of the group $S L_{2}(\mathbb{Z})$

Paper 4, Section II, G

commentLet $s=\sigma+i t$, where $\sigma$ and $t$ are real, and for $\sigma>1$ let

$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}$

Prove that $\zeta(s)$ has no zeros in the half plane $\sigma>1$. Show also that for $\sigma>1$,

$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}$

where $\mu$ denotes the Möbius function. Assuming that $\zeta(s)-\frac{1}{s-1}$ has an analytic continuation to the half plane $\sigma>0$, show that if $s=1+i t$, with $t \neq 0$, and $\zeta(s)=0$ then $s$ is at most a simple zero of $\zeta$.

1.I.1H

commentDefine the continued fraction of a real number $\alpha$.

Compute the continued fraction of $\sqrt{19}$.

2.I.1H

commentWhat does it mean for a positive definite quadratic form with integer coefficients to be reduced?

Show that there are precisely three reduced forms of this type with discriminant equal to $-23$.

Which odd primes are properly represented by some positive definite binary quadratic form (with integer coefficients) of discriminant $-23$ ?

3.I.1H

commentProve that, for all $x \geqslant 2$, we have

$\sum_{p \leqslant x} \frac{1}{p}>\log \log x-\frac{1}{2}$

[You may assume that, for $0<u<1$,

$-\log (1-u)-u<\frac{u^{2}}{2(1-u)} .$

3.II.11H

commentState the reciprocity law for the Jacobi symbol.

Let $a$ be an odd integer $>1$, which is not a square. Prove that there exists a positive integer $n$ such that $n \equiv 1 \bmod 4$ and

$\left(\frac{n}{a}\right)=-1$

Prove further that there exist infinitely many prime numbers $p$ such that

$\left(\frac{a}{p}\right)=-1$

4.I.1H

commentLet $p$ be an odd prime number. Assuming that the multiplicative group of $\mathbb{Z} / p \mathbb{Z}$ is cyclic, prove that the multiplicative group of units of $\mathbb{Z} / p^{n} \mathbb{Z}$ is cyclic for all $n \geqslant 1$.

Find an integer $a$ such that its residue class in $\mathbb{Z} / 11^{n} \mathbb{Z}$ generates the multiplicative group of units for all $n \geqslant 1$.

4.II.11H

commentLet $N>1$ be an integer, which is not a square, and let $p_{k} / q_{k}(k=1,2, \ldots)$ be the convergents to $\sqrt{N}$. Prove that

$\left|p_{k}^{2}-q_{k}^{2} N\right|<2 \sqrt{N} \quad(k=1,2, \ldots)$

Explain briefly how this result can be used to generate a factor base $B$, and a set of $B$-numbers which may lead to a factorization of $N$.

$3 . \mathrm{I} . 1 \mathrm{~F} \quad$

commentDetermine the continued fraction of $\sqrt{7}$. Deduce two pairs of solutions in positive integers $x, y$ of the equation

$x^{2}-7 y^{2}=1$

1.I.1F

commentState the prime number theorem, and Bertrand's postulate.

Let $S$ be a finite set of prime numbers, and write $f_{s}(x)$ for the number of positive integers no larger than $x$, all of whose prime factors belong to $S$. Prove that

$f_{s}(x) \leqslant 2^{\#(S)} \sqrt{x}$

where $\#(S)$ denotes the number of elements in $S$. Deduce that, if $x$ is a strictly positive integer, we have

$\pi(x) \geqslant \frac{\log x}{2 \log 2} .$

2.I.1F

commentLet $p$ be an odd prime number. Prove that 2 is a quadratic residue modulo $p$ when $p \equiv 7 \quad(\bmod 8)$. Deduce that, if $q$ is a prime number strictly greater than 3 with $q \equiv 3$ $(\bmod 4)$ such that $2 q+1$ is also a prime number, then $2^{q}-1$ is necessarily composite. Why does the argument break down for $q=3$ ?

3.II.11F

commentState the Chinese remainder theorem. Let $n$ be an odd positive integer. If $n$ is divisible by the square of a prime number $p$, prove that there exists an integer $z$ such that $z^{p} \equiv 1(\bmod n)$ but $z \not \equiv 1(\bmod n)$.

Define the Jacobi symbol

$\left(\frac{a}{n}\right)$

for any non-zero integer $a$. Give a numerical example to show that

$\left(\frac{a}{n}\right)=+1$

does not imply in general that $a$ is a square modulo $n$. State and prove the law of quadratic reciprocity for the Jacobi symbol.

[You may assume the law of quadratic reciprocity for the Legendre symbol.]

Assume now that $n$ is divisible by the square of a prime number. Prove that there exists an integer $a$ with $(a, n)=1$ such that the congruence

$a^{\frac{n-1}{2}} \equiv\left(\frac{a}{n}\right) \quad(\bmod n)$

does not hold. Show further that this congruence fails to hold for at least half of all relatively prime residue classes modulo $n$.

4.I.1F

commentProve Legendre's formula relating $\pi(x)$ and $\pi(\sqrt{x})$ for any positive real number $x$. Use this formula to compute $\pi(48)$.

4.II.11F

Let $p$ be a prime number, and let $f(x)$ be a polynomial with integer coefficients, whose leading coefficient is not divisible by $p$. Prove that the congruence

$f(x) \equiv 0 \quad(\bmod p)$

has at most $d$ solutions, where $d$ is the degree of $f(x)$.

Deduce that all coefficients of the polynomial

$x^{p-1}-1-((x-1)(x-2) \cdots(x-p+1))$