Let denote the sum of the squares of the positive integers and prime to .

Prove that

I have trouble with separating the integers into classes, so that all integers such that are in the same class.

Thanks for the help.

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- Nov 23rd 2009, 09:33 PMilikecandySums of squares of positive integers prime to n
Let denote the sum of the squares of the positive integers and prime to .

Prove that

I have trouble with separating the integers into classes, so that all integers such that are in the same class.

Thanks for the help. - Nov 24th 2009, 12:15 AMShanks
Let , for each d|n, define , then is a partion on N.

And for each , Since , By the definition of S(.), the sum of elements in is ,

Thus,the first equality of

is proved.

When d exhaust all the divisors of n, so do , Thus the second equality obviously hold.