【摘要】 丢番图方程 a2 +b2 =c2 满足条件 a >0 ,b>0 ,c>0 ,(a,b) =1的整数解 (a,b,c)称为本原勾股数 .设 x为给定的正实数 ,用 G(x)表示弦 c≤ x的所有本原勾股数 (a,b,c)的组数 .在此证明了文 [1 ]提出的本原勾股数组数 G(x)的渐近阶猜想 G(x) =1πx +O(x12 logx)的正确性 ,由此推得 limx→ +∞G(x)x =1π,即弦 c≤ x的所有本原勾股数 (a,b,c)的组数的平均阶为 1π.更多还原

【Abstract】 The integer solution (a,b,c) of the Diophantine equation a\+2+b\+2=c\+2 , which satisfies condition a>0,b>0,c>0 and (a,b) =1, is called Primitive Pythagorean Triple. Let x be a given positive real number, it is assumed that G(x) represents the group number of all the Primitive Pythagorean Triples (a,b,c) which meet c≤x . In this paper, the conjecture G(x)=1 π x+O(x\+\{12\} log x) on asymptotic order of the group number of the Primitive Pythagorean Triple proposed in\ is verified, as a result \{ lim \}x→+∞G(x)x=1 π . This means that the average order of the group number of all Primitive Pythagorean Triples (a,b,c) , which meet c≤x , is1π.更多还原