# Linear equations in primes

@article{Green2006LinearEI, title={Linear equations in primes}, author={Ben Green and Terence Tao}, journal={Annals of Mathematics}, year={2006}, volume={171}, pages={1753-1850} }

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N →∞, for the number of integer points n ∈ ℤ d ∩ K for which the integers ψ 1 (n), ... , ψ t (n) are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and…

## 383 Citations

### Diophantine equations in primes: Density of prime points on affine hypersurfaces

- MathematicsDuke Mathematical Journal
- 2022

Let F ∈ Z[x1, . . . , xn] be a homogeneous form of degree d ≥ 2, and let V ∗ F denote the singular locus of the affine variety V (F ) = {z ∈ C : F (z) = 0}. In this paper, we prove the existence of…

### LINEAR FORMS AND QUADRATIC UNIFORMITY FOR FUNCTIONS ON n p

- Mathematics
- 2011

We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176], which shows that a system…

### Linear, bilinear and polynomial structures in function fields and the primes

- Mathematics, Computer Science
- 2018

A bidirectional additive smoothing result for sets of pairs P ⊂ Fp × Fp is proved, which is a bilinear version of Bogolyubov’s theorem.

### The true complexity of a system of linear equations

- Mathematics
- 2010

In this paper we look for conditions that are sufficient to guarantee that a subset A of a finite Abelian group G contains the ‘expected’ number of linear configurations of a given type. The simplest…

### On the correlation of completely multiplicative functions

- Mathematics
- 2013

This dissertation focuses on a conjecture of S. Chowla which asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial of degree d. This…

### The primes contain arbitrarily long polynomial progressions

- Mathematics
- 2006

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1, …, Pk ∈ Z[m] in one unknown m with P1(0) = … = Pk(0) = 0,…

### Polynomials represented by norm forms via the beta sieve

- Mathematics
- 2022

A central question in Arithmetic geometry is to determine for which polynomials f ∈ Z[t] and which number fields K the Hasse principle holds for the affine equation f(t) = NK/Q(x) 6= 0. Whilst…

### PRIME SOLUTIONS TO POLYNOMIAL EQUATIONS IN MANY VARIABLES AND DIFFERING DEGREES

- MathematicsForum of Mathematics, Sigma
- 2018

Let $\mathbf{f}=(f_{1},\ldots ,f_{R})$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of…

### General systems of linear forms: equidistribution and true complexity

- MathematicsElectron. Colloquium Comput. Complex.
- 2014

### Correlations of the divisor function

- Mathematics
- 2012

Let τ(n)= ∑d 1d|n denote the divisor function. Employing the methods Green and Tao developed in their work on prime numbers, we give an asymptotic for the following correlation En∈[ −N,N…

## References

SHOWING 1-10 OF 57 REFERENCES

### The Prime k-Tuplets Conjecture on Average

- Mathematics
- 1990

The well-known twin prime conjecture states that there are infinitely many primes p such that p + 2 is also a prime. Although the proof of this seemingly simple statement is hopeless at present many…

### On the Additive Theory of Prime Numbers

- Mathematics, Computer ScienceFundam. Informaticae
- 2007

The undecidability of the additive theory of prime numbers (with identity) as well as the theory Th(N, +, n → p_n), where pn denotes the (n + 1)-th prime, are shown and r_n is the remainder of pn divided by n in the euclidian division.

### The Gaussian primes contain arbitrarily shaped constellations

- Mathematics
- 2005

We show that the Gaussian primesP[i] ⊆ ℤ[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, we show that given any distinct Gaussian integersv0,…,vk−1,…

### On sets of integers containing k elements in arithmetic progression

- Mathematics
- 1975

In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic…

### Nonconventional ergodic averages and nilmanifolds

- Mathematics
- 2005

We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions…

### A Quantitative Ergodic Theory Proof of Szemerédi's Theorem

- MathematicsElectron. J. Comb.
- 2006

A quantitative, self-contained version of this ergodic theory proof is presented, which is ``elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or theUse of the Fourier transform or inverse theorems from additive combinatorics.

### The primes contain arbitrarily long arithmetic progressions

- Mathematics
- 2004

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of…

### Progressions arithmétiques dans les nombres premiers, d'après B. Green et T. Tao

- Mathematics
- 2005

B. Green and T. Tao have recently proved that 'the set of primes contains arbitrary long arithmetic progressions', answering to an old question with a remarkably simple formulation. The proof does…

### Roth's theorem in the primes

- Mathematics
- 2003

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood…

### From combinatorics to ergodic theory and back again

- Mathematics
- 2006

Multiple ergodic averages, such as the average of expressions like f1(T nx)
f2(T 2nx) . . . fk(T knx), were first studied in the ergodic theoretic proof of Szemeredi�s Theorem
on arithmetic…