For all large x, there is a prime between x – x^0.525 and x

You need to know: Prime numbers, fractional and decimal exponents.

Background: Not needed.

The Theorem: On 3rd May 2000 Roger Baker, Glyn Harman, and János Pintz submitted to the Proceedings of the London Mathematical Society a paper in which they proved the existence of constant $x_0$ such that for all $x>x_0$ the interval $[x - x^{0.525},x]$ contains at least one prime number.

Short context: An old conjecture of Legendre (who lived in 1752-1833) states that, for every positive integer n, there is a prime number between $n^2$ and $(n+1)^2$ . This is problem 3 from famous list of Landau’s problems and remains open. It would follow from existence of primes between $x - x^\gamma$ and $x$ for $\gamma=0.5$ . However, this statement was known only for $\gamma=0.535$ . The Theorem proves it for $\gamma=0.525$ and $x>x_0$ . With enough effort, the explicit value of $x_0$ can be extracted from the proof, but this was not done.

Links: The original paper is available here.

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Every elliptic curve over Q is modular

You need to know: Basic complex analysis, infinite series, irreducible polynomial, and either “rational map” or function field.

Background: Let ${\mathbb H} = \{z \in {\mathbb C}, \text{Im}(z) > 0\}$ , $j:{\mathbb H}\to{\mathbb C}$ be given by $j(z) = 1728\frac{20 G_4(z)^3}{20 G_4(z)^3-49G_6(z)^2}$ , where $G_k(z)=\sum\limits_{(m,n)\neq (0,0)}(m+nz)^{-k}$ . For every positive integer n, there exists a non-zero irreducible polynomial $P_n(x,y)$ with integer coefficients such that $P_n(j(nz),j(z))=0, \, z\in {\mathbb H}$ . The set $X_0(n)$ of pairs of complex numbers $(x,y)$ such that $P_n(x,y)=0$ is called the classical modular curve.

Elliptic curve E over ${\mathbb Q}$ is the set of solutions to the equation $y^2=x^3+ax+b$ , where $a,b \in {\mathbb Q}$ are such that $4a^3+27b^2 \neq 0$ . It is called modular if it can be obtained via a rational map with integer coefficients from $X_0(n)$ for some positive integer n. Equivalently, E is modular if the field of functions on E (given by $Q(x)[y]/(y^2-(x^3+ax+b))$ ) is contained in the field of functions on $X_0(n)$ for some n (given by given by $Q(x)[y]/P_n(x,y)$ ).

The Theorem: On 28th February 2000 Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor submitted to the Journal of the AMS a paper in which they proved that every Elliptic curve over ${\mathbb Q}$ is modular.

Short context: The Theorem confirms conjecture of Taniyama and Shimura from 1961. In 1995, Wiles proved a special case of this conjecture and deduced Fermat’s Last Theorem, which was probably the most famous open problem in the whole mathematics for over 300 years. The Theorem confirms Taniyama-Shimura conjecture in full, has the name “the modularity theorem”, and is considered by many as one of the greatest achievements in the modern mathematics.

Links: The original paper is available here.

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The number of integer solutions to |F|<= m for decomposable form F

You need to know: Polynomial in n variables, degree of a polynomial, complex numbers.

Background: Decomposable form is a polynomial $F(x_1, \dots x_n)=\prod\limits_{i=1}^d (a_{i1}x_1+\dots+a_{in}x_n),$ where the coefficients $a_{ij}$ are non-zero complex numbers.

The Theorem: On 24th January 2000 Jeffrey Thunder submitted to Annals of Mathematics a paper in which he proved, among other results, that for every decomposable form F of degree d in n variables with integer coefficients, the number $N_F(m)$ of integer solutions to the inequality $|F(x_1, \dots, x_n)|\leq m$ is either infinite or at most $c(n,d)m^{n/d}$ , where $c(n,d)$ is an effectively computable constant depending only on n and d.

Short context: Counting integer solutions to equations and inequalities is an old theme in mathematics. Case d=2 of the Theorem was proved by Mahler in 1933, but progress in estimating $N_F(m)$ for d>2 was limited. The Theorem estimates $N_F(m)$ for all d, confirming 1989 conjecture of Schmidt.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 1.3 of this book for an accessible description of the Theorem.

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There exists a field with u-invariant 9

You need to know: The concept of a Field.

Background: A number u(F) is called u-invariant of a field F, if (a) equation $\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j = 0$ (where $x_1, x_2, \dots, x_n$ are variables and $a_{ij}$ are some coefficients from F) in n=u(F) variables may have no solutions, except of $x_1 = x_2 = \dots = x_n = 0$ , and (b) the same equation in $n=u(F)+1$ variables always have a non-zero solution.

The Theorem: On 4th January 2000 Oleg Izhboldin submitted to Annals of Mathematics a paper in which he proved that there exists a field F with u-invariant 9.

Short context: In 1953, Kaplansky conjectured that u-invariant of any field is always a power of 2. In 1989, Merkurev disproved this, and showed that u-invariant can be any even number, but left open whether it can be any odd number grater than 1. It is known that u-invariant can never be 3, 5, or 7. However, the Theorem proves that it can be 9.

Links: The original paper is available here. See also Section 1.7 of this book for an accessible description of the Theorem.

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For every n>30, n-th Lehmer number has a primitive divisor

You need to know: prime numbers, coprime integers, complex numbers, polynomials, root of a polynomial

Background: An algebraic integer is a complex number that is a root of some polynomial with integer coefficients and leading coefficient 1. A root of unity is a complex number z such that $z^n=1$ for some positive integer n. A pair of algebraic integers $(\alpha,\beta)$ is called Lehmer pair if $(\alpha+\beta)^2$ and $\alpha\beta$ are non-zero coprime integers and $\alpha/\beta$ is not a root of unity. For every Lehmer pair $(\alpha,\beta)$ , define sequence $u_n = u_n(\alpha,\beta)$ by formulas $u_n = \frac{\alpha^n - \beta^n}{\alpha-\beta}$ for odd n and $u_n = \frac{\alpha^n - \beta^n}{\alpha^2-\beta^2}$ for even n. Then $u_n, \, n=0,1,,2,\dots$ is a sequence of integers which is called the sequence of Lehmer numbers. A prime number p is called a primitive divisor of $u_n$ if p divides $u_n$ but does not divide $(\alpha^2-\beta^2)^2 u_1 \dots u_{n-1}$ .

The Theorem: On 9th February 1999, Yuri Bilu, Guillaume Hanrot and Paul Voutier submitted to Journal fur die Reine und Angewandte Mathematik a paper in which they proved that for every Lehmer pair $(\alpha,\beta)$ and every n>30, $u_n(\alpha,\beta)$ has a primitive divisor.

Short context: The condition n>30 in the Theorem is the best possible because, for example, $\alpha=(1-\sqrt{-7})/2$ and $\beta=(1+\sqrt{-7})/2$ is a Lehmer pair for which $u_{30}$ has no primitive divisors. In fact, an old and difficult problem, which goes back to at least the beginning of 20-th century, is to find all triples $(\alpha,\beta, n)$ such that $(\alpha,\beta)$ is a Lehmer pair and $u_n(\alpha,\beta)$ has no primitive divisors. With the help of computer, the authors found all such triples with $n\leq 30$ , while the Theorem states that there are no such triples with $n>30$ . This completely solves the problem. As a special case, the Theorem also solves the corresponding problem for so-called Lucas numbers.

Links: The original paper is here.

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The asymptotic mean of bounded muptiplicative function is at least -0.656999…

You need to know: limits, integrals, logarithm $\log$ , base of natural logarithms e.

Background: Let ${\mathbb N}$ be the set of positive integers and ${\mathbb R}$ be the real line. A function $f:{\mathbb N} \to {\mathbb R}$ is called completely multiplicative, if $f(x \cdot y) = f(x) \cdot f(y)$ for all positive integers x, y. We say that function $g:{\mathbb N} \to {\mathbb R}$ is $o(1)$ if $\lim\limits_{n\to\infty} g(n) =0$ .

The Theorem: On 8th September 1999, Andrew Granville and Kannan Soundararajan submitted to Annals of Mathematics a paper in which they proved that for every completely multiplicative function $f:{\mathbb N} \to {\mathbb R}$ , taking values in $[-1,1]$ , $\frac{1}{N}\sum\limits_{n=1}^N f(n) \geq \delta + o(1)$ , where $\delta = 1-2\log(1+\sqrt{e})+4\int_1^{\sqrt{e}} \frac{\log t}{t+1}dt = -0.656999...$ .

Short context: In 1994, Heath-Brown conjectured that $\frac{1}{N}\sum\limits_{n=1}^N f(n) \geq c + o(1)$ for some constant $c > -1$ . In 1996, Hall proved this conjecture, and in turn conjectured that one can take $c=\delta$ with $\delta$ defined above. The Theorem confirms this conjecture. As observed by Hall, this result is the best possible because there exists f for which the equality holds.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 1.2 of this book for an accessible description of the Theorem.

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There are infinitely many primes of the form x^3+2y^3

You need to know: Prime numbers.

The Theorem: On 29th April 1999, Rodger Heath-Brown submitted to Acta Mathematica a paper in which he proved that there are infinitely many primes of the form $x^3+2y^3$ with integer x, y.

Short context: Let P be a polynomial with integer coefficients in one or more variables. If we substitute integer values instead of variables, will we get infinitely many primes as values of P? This problem is wide open even for simple polynomials like $P(x)=x^2+1$ . Famous Dirichlet’s theorem solves it for linear polynomials, Iwaniec in 1974 resolved the case of quadratic polynomials in 2 variables (which depends essentially on both variables), while Friedlander and Iwaniec proved in 1998 that there are infinitely many primes of the form $x^2+y^4$ with integer x, y. The Theorem resolves this question for polynomial $P(x,y)=x^3+2y^3$ .

Links: The original paper is here.