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Problem 748


Problem 748


Upside down Diophantine equation

Upside Down is a modification of the famous Pythagorean equation:
1x2+1y2=13z2

A solution (x,y,z) to this equation with x,y and z positive integers is a primitive solution if gcd(x,y,z)=1.

Let S(N) be the sum of x+y+z over primitive Upside Down solutions such that 1x,y,zN and xy.
For N=100 the primitive solutions are (2,3,6) and (5,90,18), thus S(102)=124.
It can be checked that S(103)=1470 and S(105)=2340084.

Find S(1016) and give the last 9 digits as your answer.


颠倒不定方程

把著名的毕达哥拉斯方程稍加修改就得到了如下的颠倒不定方程:
1x2+1y2=13z2

若上述方程的一组正整数解(x,y,z)满足gcd(x,y,z)=1,则称之为该方程的本原解。

考虑该方程的所有本原解,并记S(N)为所有满足1x,y,zNxy的本原解对应x+y+z之和。
对于N=100,满足条件的本原解包括(2,3,6)(5,90,18),因此S(102)=124
可以验证,S(103)=1470S(105)=2340084

S(1016),并给出最后9位数字作为你的答案。


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