算术级数中的五素数平方和定理

On Hua's Five Prime Squares Theorem with One Prime in Arithmetic Progressions

设N是充分大的正整数满足N≡5mod 24,l和d是满足(l,d)=1的整数.A0,A>1是满足A0=600A+2000的正常数.本文证明对所有的整数0<d≤D0= N^(1/4)log^(-A0)N,除了至多O(D0log-AN)个例外,方程N=p12+p22+…+p52有素数解p1,p2,…,p5,其中p1≈l mod d.

Let N be a sufficiently large positive integer satisfying N ≡ 5 mod 24, l and d be integers satisfying （l, d） = 1. Denote by A0 and A 〉 1 the positive constants, satisfying A0 = 600A＋2000. For all integers 0 〈 d ≤ D0 = N^1/4 log^-A0 N, with at most O（D0 log^-A N） exceptional values, the equation N = p1^2 ＋ p2^2 ＋ … ＋ p5^2 has solutions in primes p1,p2,…,p5, such that p1 ≡ l mod d.