Write , with . Show that divides a if

I have no clue how to express it as .

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- Nov 8th 2009, 09:54 AMcomssaSum of reciprocals of phi(p)
Write , with . Show that divides a if

I have no clue how to express it as . - Nov 8th 2009, 02:49 PMtonio

I can't understand what you think you can see here: , since, for example, .

Anyway, and since , for simplicity of the proof we'll prove that in fact

Now, we can form pairs , so

, so we have to prove this last sum is divisible by p (the minus sign never minds: we can take it out of the sum).

But as the only elements in which are inverses to themselves are 1 and -1 and without these two we're left with an even number of

elements, from 2 to p-2 (and here enters the assumption ), where we can pair each one with its inverse. Now, the well-known formula

means the sum equals zero modulo p since, again, p is prime greater than 5. Q.E.D.

Tonio