# Lagrange's four squares theorem with one prime and threealmost--prime variables

@article{HeathBrown2003LagrangesFS, title={Lagrange's four squares theorem with one prime and threealmost--prime variables}, author={D. R. Heath-Brown and D. Tolev}, journal={Crelle's Journal}, year={2003}, volume={2003}, pages={159-224} }

It is conjectured that every sufficiently large integer $N\equiv 4\pmod{24}$ should be a sum of the squares of 4 primes. The best approximation to this in the literature is the result of Brudern and Fouvry [J. Reine Angew. Math., 454 (1994), 59--96] who showed that every sufficiently large integer $N\equiv 4\pmod{24}$ is a sum of the squares of 4 almost-primes, each of which has at most 34 prime factors.
The present paper proves such a result with the square of one prime and 3 almost-primes… Expand

#### 19 Citations

LAGRANGE'S FOUR SQUARES THEOREM WITH VARIABLES OF SPECIAL TYPE

- Mathematics
- 2010

Let N denote a sufficiently large integer satisfying N ≡ 4 (mod 24), and Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that the… Expand

Lagrange's equation with one prime and three almost-primes

- Mathematics
- 2018

Abstract In this paper, we consider the representation of a large positive integer N ≡ 4 ( mod 24 ) in the form p 2 + x 1 2 + x 2 2 + x 3 2 where p is a prime number and x 1 , x 2 , x 3 are… Expand

Lagrange's equation with almost-prime variables

- Mathematics
- 2020

Abstract We investigate Lagrange's equation with almost-prime variables. We establish the result that every sufficiently large integer of the form 24 k + 4 can be represented as the sum of four… Expand

Sarnak's saturation problem for complete intersections

- Mathematics
- 2017

We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size B with each component having no prime divisors… Expand

On the slim exceptional set for the Lagrange four squares theorem

- Mathematics
- 2012

Abstract.Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E3(N) denote the number of natural numbers not exceeding N that are congruent to 4… Expand

On a generalization of Hua’s theorem with five squares of primes

- Mathematics
- 2009

AbstractWe sharpen Hua’s theorem with five squares of primes by proving that every sufficiently large integer N congruent to 5 modulo 24 can be written in the form $$
N = p_1^2 + p_2^2 + p_3^2 +… Expand

The quadratic Waring–Goldbach problem

- Mathematics
- 2004

Abstract It is conjectured that Lagrange's theorem of four squares is true for prime variables, i.e. all positive integers n with n≡4 ( mod 24) are the sum of four squares of primes. In this paper,… Expand

On a binary Diophantine inequality involving prime numbers

- Mathematics
- 2020

Let $$1< c < \frac{59}{44},\, c\ne \frac{4}{3}$$ 1 < c < 59 44 , c ≠ 4 3 . In this paper it is proved that for any sufficiently large real N , for almost all real $$T\in (N,2N]$$ T ∈ ( N , 2 N ] (in… Expand

Sums of smooth squares

- Mathematics
- Compositio Mathematica
- 2009

Abstract Let R(n,θ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding nθ/2. When θ>e−1/3 a lower bound for R(n,θ)… Expand

A Three Squares Theorem with almost Primes

- Mathematics
- 2005

As an application of the vector sieve and uniform estimates on the Fourier coefficients of cusp forms of half-integral weight, it is shown that any sufficiently large number n ≡ 3 (mod 24) with 5 n… Expand

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