2018/11/20 by Wu, Hai-Liang, Sun, Zhi-Wei
#11A41 #11D85 #11E20 #11E25 #11F27 #11F37 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1811.08341
The well-known Lagrange's four-square theorem states that any integer n∈ℕ=\0,1,2,...\ can be written as the sum of four squares. Recently, Z.-W. Sun investigated the representations of n as x2+y2+z2+w2 with certain linear restrictions involving the integer variables x,y,z,w. In this paper, via the theory of quadratic forms, we further study the representations n=x2+y2+z2+w2 (resp., n=x2+y2+z2+2w2) with certain linear restrictions involving primes. For example, we obtain the following results: (i) Each positive integer n>1 can be written as x2+y2+z2+2w2 (x,y,z,w∈\mathbb N) with x+y prime. (ii) Every positive integer can be written as x2+y2+z2+2w2 (x,y,z,w∈\mathbb N) with x+2y prime. (iii) Let k be any positive integer, and let d≥ 2k-1 be a positive odd integer with 4d2+1 prime. Then any sufficiently large integer can be written as x2+y2+z2+2w2 (x,y,z,w∈\mathbb N) with x+2dy=pk for some prime p.