丢番图方程a^2+Db^2=c^2本原解组数的渐近阶(Ⅱ)

The asymptotic order on the group number of primitive solution of Diophantine equation a^2+Db^2=c^2(Ⅱ)

设x为给定的正实数,D是给定的正整数且无平方因子,用G(D,x)表示丢番图方程a2+Db2=c2满足条件a>0,b>0,c>0,(a,b)=1且c≤x的所有整数解(a,b,c)的组数.在此考虑D=P和D=2P(其中P=p1p2…pk为互异的奇素数的乘积)的情形,得到渐近估计式G(P,x)=d(P)Pσ(P)πx+Ox12logx和G(2P,x)=d(P)2P2σ(P)πx+Ox12logx.

Let \$x\$ be a given positive real number, \$D\$ be a given positive integer and square free, it is assumed that \$G(D,x)\$ represents the group number of all the integer solution \$(a,b,c)\$ of the Diophantine equation \$a\+2+Db\+2=c\+2\$, which satisfies condition \$a>0,b>0,c> 0,(a,b)=1\$ and \$c≤x\$.In this paper, let \$P=p\-1p\-2\:p\-k\$ be product of different odd prime numbers, it is verified that the two asymptotic estimation formulas \$G(P,x)=d(P)Pσ(P)\%π\%x+O(x\+\{12\}\%log\%x)\$ and \$G(2P,x)=d(P)2P2σ(P)\%π\%x+O(x\+\{12\}\%log\%x)\$.\;