# 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}
}
• Published 2003
• Mathematics
• Crelle's Journal
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
LAGRANGE'S FOUR SQUARES THEOREM WITH VARIABLES OF SPECIAL TYPE
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 theExpand
Lagrange's equation with one prime and three almost-primes
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 areExpand
Lagrange's equation with almost-prime variables
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 fourExpand
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 divisorsExpand
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 4Expand
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 ] (inExpand
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 nExpand