素数变数的线性方程组

<正> 引言 在苹雁庚教授的著作“堆曼素数箫”第十二章中曹握提出了阴龄整保数素数燮数的腺性方程粗的解的问题.这个问题是有名的(?)定理的自然推广.1937年苏联(?)院士首先证明了任何充分大的奇整数 N 都能表成三个素数之和,且如令 I(N) 为表示法的种数,则

ON THE SOLUTIONS OF THE SYSTEMS OF LINEAR EQUATIONS WITH PRIME VARIABLES

Let α_(11),…,α_(1,2n+1),…,α_(n1),…,α_(n,2n+1),b_1,…,b_n,be 2n(n+1)integers,and P be a sufficiently large integer.My purpose is trying to find anasymptotic formula for the number of prime number solutions of the systemof equations(?)(μ=1,…,n),within the region 2≤ρ_v≤ρ(v=1,…,2n+1)under some conditions.Thisproblem was proposed by Prof.Hua in his book“Theory of Additive Primenumbers”,and is the natural extension of the famous“Goldbach-Vinogradov's”Theorem.Putting L=logP,e(x)=e~(2πix),(Φ(p))~(2n+1)s(p)/p~n be the number of solu-tions of(?)(μ=1,…,n)within the range 1≤l_v≤ρ-1,thenⅠproved in this paper the Theorem 1.If all the n-th minors of the matrix(?)is not equal to zero,and have no common divisor other thanⅠ,then thenumber of systems of prime number solutions of the equations(?)(μ=1,…,n),within the region 2≤ρ_v≤ρ(1≤v≤2n+1)is equal to(?)where the constant in the symbol O is independent of b's,R=(logL)~n or1 when n is greater than or equal to 1,and(?)(?)where(?)Without any difficulty,by the same method as in the proof of Theorem1, we can proveTheorem 1'.Let m≥2n+1,if all the n-th minors of the matrixis not equal to zero,and have no common divisor other than 1,then(?)where I_1(b;P),B_1(b;P),s_1(p) have the same meaning as in Theorem 1,andthe constant involved in the symbol O is also independent of b's. When n=1,Ⅰprove the followingTheorem 2.Let(α_1,…,α_m)=1,m≥3,b≡sum from v=1 to m α_v(mod 2).If(α_v_1,…,α_v_(m-1),b)=1.is always true for any m—1 different α_v,then the number of prime num-ber solutions of the equationα_1p_1+…+α_mp_m=bwithin the range 2≤P_v≤P,(1≤v≤m),is equal to(?)where(?)(b)is an infinite product and is greater than an absolute constant.Moreover,if α_v>0 for all v,then(?)where A_m denotes the product α_1…α_m.