ON THE SOLUTIONS OF THE SYSTEMS OF LINEAR EQUATIONS WITH PRIME VARIABLES
ON THE SOLUTIONS OF THE SYSTEMS OF LINEAR EQUATIONS WITH PRIME VARIABLES
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发表时间：
1957
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通讯作者：
F. Wu
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作者：
F. Wu
设α_（11），.，α_（1，2n+1），.，α_（n1），.，α_（n，2n+1），b_1，.，b_n，是2n（n+1）个整数，P是一个充分大的整数，本文的目的是寻找方程组（？）（μ= 1，..，n）在一定条件下，在2≤ρ_v≤ρ（v= 1，..，2n +1）的范围内，求出了（μ= 1，..，n）的解的个数，这个问题是华国锋教授在《可加素数论》中提出的，是著名的“Goldbach-Vinogradov定理”的自然推广。（μ= 1，..，n）在1≤l_v≤ρ-1范围内，则本文证明了定理1.若矩阵（？）不等于零，且除Ⅰ外无公约数，则方程（？）（μ= 1，.，n），在2≤ρ_v≤ρ（1≤v≤2n+1）范围内，（？）其中符号O中的常数与B无关，当n大于或等于1时，R=（logL）~n或1，（？）(?) Where（？）用与定理1相同的证明方法，我们可以毫无困难地证明定理1，设m≥2n+1，若矩阵的所有n阶子式不等于零，且除1外没有公约数，则（？）其中I_1（B;P），B_1（B;P），s_1（p）的意义与定理1相同，符号O中的常数也与B无关。当n=1时，证明了定理2.设（α_1，.，α_m）=1，m≥3，B是v=1到m α_v（mod 2）的和，若（α_v_1，.，α_v_（m-1），B）= 1，对任意m-1个不同的α_v总是成立，则方程α_1p_1+.+α_mp_m= b在2≤P_v≤P，（1≤v≤m）范围内的素数解的个数等于（？）Where（？）(b)is而且，若α_v0对所有v，则（？）其中A_m表示乘积α_1…α_m。
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_v0 for all v,then(?)where A_m denotes the product α_1…α_m.