The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144

You need to know: Basic arithmetic only.

Background: Let $F_0, F_1, \dots, F_n, \dots$ be the Fibonacci sequence defined by $F_0 = 0, F_1 = 1,$ and $F_{n+2} = F_{n+1} + F_n$ for $n\geq 0$ . Let $L_0, L_1, \dots, L_n, \dots$ be the Lucas sequence defined by $L_0 = 2, L_1 = 1,$ and $L_{n+2} = L_{n+1} + L_n$ for $n\geq 0$ . A perfect power is an integer of the form $m^p$ for integers $m$ and $p\geq 2$ .

The Theorem: On 24th November 2003, Yann Bugeaud, Maurice Mignotte, and Samir Siksek submitted to the Annals of Mathematics a paper in which they proved that (a) the only perfect powers in the Fibonacci sequence are $F_0 = 0$ , $F_1 = 1$ , $F_2 = 1$ , $F_6 = 8$ , and $F_{12} = 144$ , and (b) the only perfect powers in the Lucas sequence are $L_1=1$ and $L_3=4$ .

Short context: The Fibonacci sequence is perhaps the most famous and well-studied sequence of integers in mathematics. In 1951, Ljunggren proved that the only perfect squares in this sequence are $0=0^2, 1=1^2$ , and $144=12^2$ . In 1969, London and Finkelstein proved that the only perfect cubes are $0,1$ and $8=2^3$ . Are there any other perfect powers in the sequence? By 2003, it was known that there are no more p-th powers with $p\leq 17$ or with $p\geq 5.1\cdot 10^{17}$ . The Theorem proves this for all p, and resolves this question also for the Lucas sequence.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.3 of this book for an accessible description of the Theorem.

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Two values for the chromatic number of a random graph G(n,d/n)

You need to know: Graph, vertices, edges, chromatic number of a graph, limits, basic probability theory, independent events.

Background: Let $G(n,p)$ denote random graph with n vertices, such that every possible edge is included independently with probability p. For a fixed $d>0$ , consider sequence of random graphs $G\left(n,\frac{d}{n}\right)$ with $n\to\infty$ .

The Theorem: On 9th October 2003, Dimitris Achlioptas and Assaf Naor submitted to the Annals of Mathematics a paper in which they proved that, with probability that tends to 1 as $n\to\infty$ , the chromatic number of $G\left(n,\frac{d}{n}\right)$ is either $k_d$ or $k_d+1$ , where $k_d$ denotes the smallest integer k such that $d<2k\log k$ .

Short context: Chromatic number is one of the most important and well-studied invariants of a graph, and it is natural to ask what it is equal to for random graphs. In 1991, Luczak proved that for every $d>0$ there exists a constant $k_d$ , such that, with probability that tends to 1 as $n\to\infty$ , the chromatic number of $G\left(n,\frac{d}{n}\right)$ is either $k_d$ or $k_d+1$ . However, the exact value of $k_d$ remained unknown. The Theorem answers this question.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 5.11 of this book for an accessible description of the Theorem.

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Lehmer’s conjecture is true for polynomials with odd coefficients

You need to know: Polynomials, degree of a polynomial, roots of a polynomial, divisor of a polynomial, irreducible polynomial, complex numbers, notation ${\mathbb Z}[x]$ for polynomials in variable x with integer coefficients, notation $a \equiv b\,(\text{mod}\, m)$ if $a-b$ is divisible by m.

Background: An irreducible polynomial $P \in {\mathbb Z}[x]$ is cyclotomic if $P$ is a divisor of $x^n-1$ for some integer $n \geq 1$ . Every polynomial $P \in {\mathbb Z}[x]$ of degree n can be written as $P(x)=a\prod\limits_{i=1}^n(x-\alpha_i)$ , where $a\in {\mathbb Z}$ and $\alpha_i$ are (possibly complex) roots of P. Mahler ’s measure of P is $M(P):=|a|\prod\limits_{i=1}^n\max\{1,|\alpha_i|\}$ .

For integer $m\geq 2$ , let $D_m:=\left\{\sum\limits_{i=0}^n a_ix^i \in {\mathbb Z}[x]\,|\,a_i \equiv 1 (\text{mod}\, m), \, 0\leq i \leq n\right\}$ .

The Theorem: On 2nd October 2003, Peter Borwein, Edward Dobrowolski, and Michael Mossinghoff submitted to the Annals of Mathematics a paper in which they proved that inequality $\log M(P) \geq c_m\left(1-\frac{1}{n+1}\right)$ holds for every $P \in D_m$ of degree n and no cyclotomic divisors, where $c_2=(\log 5)/4$ and $c_m=\log(\sqrt{m^2+1}/2)$ for $m>2$ .

Short context: In 1933, Lehmer asked if for every $\epsilon>0$ there exists a polynomial $P \in {\mathbb Z}[x]$ satisfying $1<M(P)<1+\epsilon$ . It is conjectured that the answer to this question is negative, and this is known as Lehmer’s conjecture. The Theorem implies that this conjecture holds for polynomials $P \in D_m$ . In particular, case $m=2$ of the Theorem implies that Lehmer’s conjecture holds for polynomials with odd coefficients.

Links: The original paper is available here. See also Section 7.7 of this book for an accessible description of the Theorem.

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The kissing number in dimension four is 24

You need to know: n-dimensional Euclidean space ${\mathbb R}^n$ , sphere in ${\mathbb R}^n$ , nonoverlapping spheres, touching spheres.

Background: The kissing number $k(n)$ is the highest number of equal nonoverlapping spheres in ${\mathbb R}^n$ that can touch another sphere of the same size.

The Theorem: On 26th September 2003, Oleg Musin submitted to arxiv a paper in which he proved that $k(4)=24$ .

Short context: It is easy to see that $k(1)=2$ and $k(2)=6$ . Determining $k(3)$ (the maximal number of white billiard balls that can touch a black ball) is an old problem which goes back to at least 1694. In 1953, Schütte and van der Waerden proved that $k(3)=12$ . In 1979, Levenshtein, and independently Odlyzko and Sloane, proved that $k(8) = 240$ , and $k(24) = 196560$ , but, before 2003, the problem was open in all other dimensions. In ${\mathbb R}^4$ , it is easy to see that $k(4) \geq 24$ . In 1978, Böröczky proved a conjecture of Coxeter which implies that $k(4) \leq 26$ . In 1979, Odlyzko and Sloane proved that $k(4) \leq 25$ . The Theorem proves that $k(4)<25$ , which implies that $k(4)=24$ .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 8.8 of this book for an accessible description of the Theorem.

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The Trudinger-Moser inequality with Sobolev norm holds for all domains in R^2

You need to know: Euclidean space ${\mathbb R}^2$ of pairs $x=(x_1, x_2)$ with norm $|x|=\sqrt{x_1^2+x_2^2}$ , differentiable function on ${\mathbb R}^2$ , its gradient $\nabla u(x)=\left(\frac{\partial u}{\partial x_1}(x), \frac{\partial u}{\partial x_2}(x) \right)$ , integral over $\Omega \subset {\mathbb R}^2$ .

Background: Let $\Omega \subset {\mathbb R}^2$ . The Sobolev norm of a differentiable function $u:\Omega \to {\mathbb R}$ is $||u||_S := \left(\int_\Omega\left(|\nabla u(x)|^2 + |u(x)|^2\right)dx\right)^{1/2}.$

The Theorem: On 25th September 2003, Berdhard Ruf submitted to the Journal of Functional Analysis a paper in which he proved the existence of constant $d > 0$ such that for any $\Omega \subset {\mathbb R}^2$ , $\sup\limits_{||u||_S\leq 1}\int_\Omega \left(e^{4\pi u^2}-1\right)dx \leq d.$

Short context: Let $\Omega \subset {\mathbb R}^2$ be bounded, and let $||u||_D := \left(\int_\Omega\left(|\nabla u(x)|^2 \right)dx\right)^{1/2}$ denote the Dirichlet norm of function $u:\Omega \to {\mathbb R}$ . In analysis, it is important to understand what is the fastest-growing function g such that $\sup\limits_{||u||_D\leq 1}\int_\Omega g(u(x)) dx < \infty$ . The answer is given by classical Trudinger-Moser inequality, stating that $\sup\limits_{||u||_D\leq 1}\int_\Omega \left(e^{\alpha u^2}-1\right) dx = c(\Omega)< \infty$ for $\alpha\leq 4\pi$ , but the supremum becomes infinite for $\alpha>4\pi$ . However, constant $c(\Omega)$ depends on the domain, grows to infinity if area of $\Omega$ grows, and becomes infinite for unbounded domains. The Theorem proves that with the Sobolev norm in place of the Dirichlet norm, the Trudinger-Moser inequality holds with constant independent of $\Omega$ , and therefore is valid for all domains, including the unbounded ones.

Links: The original paper is available here.

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Any C^(3) nondegenerate planar curve is of Khintchine type for divergence

You need to know: Differentiable function, (k times) continuously differentiable function, sets of measure $0$ in ${\mathbb R}$ and ${\mathbb R}^2$ , notion of “almost all”, sum of infinite series, standard notations ${\mathbb R}^+$ for positive real numbers, ${\mathbb Z}$ for integers, ${\mathbb N}$ for natural numbers, $f'(x)$ for derivative, $f''(x)$ for the second derivative.

Background: Let $\psi: {\mathbb R}^+ \to {\mathbb R}^+$ be a decreasing function. Pair $(\alpha,\beta) \in {\mathbb R}^2$ is called simultaneously $\psi$ -approximable if there are infinitely many $n \in {\mathbb N}$ such that $\max\{||n\alpha||,||n\beta||\} < \psi(n)$ , where $||x||=\min\{|x-m|: m \in {\mathbb Z}\}$ denotes the distance from x to the nearest integer.

The Theorem: On 24th September 2003, Victor Beresnevich, Detta Dickinson, and Sanju Velani submitted to the Annals of Mathematics a paper in which they proved the following result. Let $\psi: {\mathbb R}^+ \to {\mathbb R}^+$ be a decreasing function with $\sum\limits_{n=1}^\infty \psi(n)^2 = \infty$ . Let f be a three times continuously differentiable function on some interval $(a, b)$ , such that $f''(x) \neq 0$ for almost all $x\in(a,b)$ . Then for almost all $x\in(a,b)$ the pair $(x, f(x))$ is simultaneously $\psi$ -approximable.

Short context: Let $S(\psi)$ be the set of all pairs $(\alpha,\beta) \in {\mathbb R}^2$ that are simultaneously $\psi$ -approximable. The (two-dimensional version of) famous Dirichlet’s approximation theorem implies that $S(\psi)={\mathbb R}^2$ for $\phi(n)=\frac{1}{\sqrt{n}}$ (see here for the progress about related conjecture of Littlewood). In 1924, Khintchin proved that almost all pairs $(\alpha,\beta) \in {\mathbb R}^2$ belong to $S(\psi)$ if and only if $\sum\limits_{n=1}^\infty \psi(n)^2 = \infty$ . However, planar curves of the form $(x, f(x))$ has measure $0$ in ${\mathbb R}^2$ , so Khintchin’s theorem says nothing about simultaneous approximability on such curves. A curve is called of Khintchine type for divergence if almost all points on it belong to $S(\psi)$ whenever $\sum\limits_{n=1}^\infty \psi(n)^2 = \infty$ . The Theorem proves this property for a broad class of curves (which are called $C^{(3)}$ nondegenerate planar curves).

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.8 of this book for an accessible description of the Theorem.

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Whitney extension theorem holds in sharp form

You need to know: Polynomials and functions on n variables, higher order partial derivatives, Lipschitz function.

Background: For a function $f:{\mathbb R}^n\to{\mathbb R}$ and vector $\beta=(\beta_1, \beta_2 \dots, \beta_n)$ of non-negative integers, denote $\partial^\beta f$ the partial derivative $\frac{\partial^{|\beta|}f}{\partial x_1^{\beta_1} \partial x_2^{\beta_2} \dots \partial x_n^{\beta_n}}$ of order $|\beta|=\beta_1 + \beta_2 + \dots + \beta_n$ . Let $C^m({\mathbb R}^n)$ denote the space of functions $f:{\mathbb R}^n\to{\mathbb R}$ whose derivatives of order $\leq m$ are continuous and bounded on ${\mathbb R}^n$ . Also, let $C^{m-1,1}({\mathbb R}^n)$ denote the space of functions whose derivatives of order $m-1$ are Lipschitz with constant 1.

The Theorem: On 14th May 2003, Charles Fefferman submitted to the Annals of Mathematics a paper in which he proved that for any integers $m\geq 1$ and $n\geq 1$ there exists an integer k, depending only on m and n, for which the following holds. Let $f:E\to{\mathbb R}$ be a function defined on an arbitrary subset $E$ of ${\mathbb R}^n$ . Suppose that, for any k distinct points $x_1,\dots, x_k \in E$ there exist polynomials $P_1,\dots, P_k$ on ${\mathbb R}^n$ of degree $m-1$ satisfying (a) $P_i(x_i)=f(x_i)$ for $i=1,\dots, k$ ; (b) $|\partial^\beta P_i(x_i)| \leq\ M$ for $i=1,\dots, k$ and $|\beta|\leq m-1$ ; and (c) $|\partial^\beta (P_i-P_j)(x_i)| \leq M|x_i-x_j|^{m-|\beta|}$ for $i,j=1,\dots, k$ and $|\beta|\leq m-1$ , where M is a constant independent of $x_1,\dots, x_k$ . Then $f$ extends to $C^{m-1,1}$ function on ${\mathbb R}^n$ .

Short context: Given a function $f:E\to{\mathbb R}$ , where E is a subset of ${\mathbb R}^n$ , how can we decide whether f extends to a $C^m({\mathbb R}^n)$ function F on ${\mathbb R}^n$ ? This is a classical question which has been answered by Whitney in 1934, and the result is known as Whitney extension theorem. The Theorem solves a version of this problem in which F is required to be $C^{m-1,1}({\mathbb R}^n)$ .

Links: The original paper is available here. See also Section 5.2 of this book for an accessible description of the Theorem.

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Every separable infinite dimensional Banach space has infinite diameter

You need to know: Bijection, infimum, supremum, Banach space B, notation $\|.\|_B$ for norm in B, dimension of a Banach space, infinite-dimensional Banach space.

Background: A Banach space B is called separable if it contains a sequence $x_1, x_2,\dots, x_n,\dots$ such that for any $x \in B$ and any $\epsilon>0$ there exists an n such that $||x-x_n||_B<\epsilon$ . Banach spaces $B$ and $B'$ are isomorphic if there exist a bijection $f:B\to B'$ , which preserves addition and multiplication by constants, and constants $m>0$ and $M>0$ such that $m\|x\|_B\leq \|f(x)\|_{B'} \leq M\|x\|_B, \, \forall x\in B$ . Assuming that constants m and M are chosen to be best possible, we denote by $d_f(B,B')$ the ratio $M/m$ . Let $d(B,B')$ be the infimum of $d_f(B,B')$ over all f satisfying the conditions above. Finally, diameter $D(B)$ of the Banach space $B$ is the supremum of $d(B',B'')$ over all Banach spaces $B'$ and $B''$ isomorphic to $B$ .

The Theorem: On 23rd September 2003, William Johnson and Edward Odell submitted to the Annals of Mathematics a paper in which they proved that if B is a separable infinite-dimensional Banach space, then $D(B)=\infty$ .

Short context: In 1981, Gluskin proved the existence of constant $c>0$ such that inequality $cN \leq D(B)$ holds for every Banach space $B$ of finite dimension N (it is also known that $D(B) \leq N$ ). From this, it is natural to conjecture that if dimension N is infinite, then $D(B)$ should be infinite as well. In fact, this problem was raised by Schäffer in 1976, even before the Gluskin’s result. The Theorem resolves this problem for separable Banach spaces.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 5.6 of this book for an accessible description of the Theorem.

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The Hopf condition for bilinear forms holds over arbitrary fields

You need to know: Factorial $n!$ of a non-negative integer n. Field.

Background: We say that field F with additive identity $0$ and multiplicative identity $1$ has characteristic not equal to 2 if $1+1 \neq 0$ . For a field F and positive integers r, s, and n, by sums-of-squares formula of type [r, s, n] over F we mean an identity of the form $\left(\sum\limits_{i=1}^r x_i^2\right)\cdot \left(\sum\limits_{i=1}^s y_i^2\right) = \sum\limits_{i=1}^n z_i^2$ , where each $z_i$ is a bilinear form, that is, expression of the form $z_i=\sum\limits_{j=1}^r\sum\limits_{k=1}^s c_{ijk}x_jy_k$ , with some coefficients $c_{ijk}\in F$ .

The Theorem: On 11th September 2003, Daniel Dugger and Daniel Isaksen submitted to arxiv a paper in which they proved that, if F is a field of characteristic not equal to 2, and a sums-of-squares formula of type [r, s, n] exists over F, then the numbers $\frac{n!}{i!(n-i )!}$ are even integers for all i such that $n-r < i < s$ .

Short context: For the special case when $F={\mathbb Q}$ is the field of rational numbers, the Theorem has been known since 1939, and the condition “the numbers $\frac{n!}{i!(n-i )!}$ are even integers for all i such that $n-r < i < s$ ” is known as the Hopf condition. This condition is one of the central tools in the problem of understanding for which r, s, and n, a sums-of-squares formula of type [r, s, n] exists. The Theorem establishes this condition over an arbitrary field (except of fields of characteristic 2 in which sums-of-squares formulas of type [r, s, n] trivially exist for all r, s, n).

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.5 of this book for an accessible description of the Theorem.

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Upper bound for the number of number fields with fixed degree and bounded discriminant

You need to know: Field, field ${\mathbb Q}$ of rational numbers, isomorphic fields, vector space, vector space over a field, dimension of a vector space, matrix, determinant of a matrix.

Background: Number field is a field F that contains ${\mathbb Q}$ and has finite dimension n when considered as a vector space over ${\mathbb Q}$ . Number n is called the degree of F. A set $e=\{e_1, e_2, \dots, e_n\}$ of n elements of F is called basis of F if every $x\in F$ can be written as $x=\sum_{i=1}^n c_i(x) e_i$ with coefficients $c_i(x) \in {\mathbb Q}$ . Sum $\text{Tr}(x)=\sum_{i=1}^nc_i(x\cdot e_i)$ does not depend on the choice of basis e and is called trace of x. If, for every $x\in F$ , all $c_i(x)$ are integers, e is called integral basis of F. The determinant of an $n\times n$ matrix with entries $\text{Tr}(e_i\cdot e_j)$ , $i=1,\dots,n$ , $j=1,\dots,n$ , does not depend on the choice of integral basis e and is called the discriminant of F. For $X>0$ , let $N_n(X)$ denotes the number of non-isomorphic number fields of degree n with absolute value of the discriminant at most X.

The Theorem: On 8th September 2003, Jordan Ellenberg and Akshay Venkatesh submitted to arxiv a paper in which they proved the existence of constant $B_n$ depending only on n and absolute constant C, such that inequality $N_n(X) \leq B_n X^{\exp(C\sqrt{\log n})}$ holds for all $n>2$ and all $X>0$ .

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is a conjecture that $N_n(X)$ grows as linear function of X for every fixed n, but, before 2003, this was known only for $n\leq 3$ (in later works – see here and here – Bhargava proved it for $n=4$ and $n=5$ ). For general n, the best upper bound was $N_n(X) \leq B_n X^{(n+2)/4}$ . The Theorem proves a bound which is significantly better for large n.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.2 of this book for an accessible description of the Theorem.

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