Hi, I've to solve the following diophantine equation (in positive integers):

.

Thanks for any help.

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- Jul 6th 2010, 08:01 PMArczi1984Diophantine equation
Hi, I've to solve the following diophantine equation (in positive integers):

.

Thanks for any help. - Jul 6th 2010, 08:34 PMAlso sprach Zarathustra
Maybe if we look the diophantine equation above .

It seems (only in theory!) that the above diophantine equation had no solution in positive integers...

http://i48.tinypic.com/vopixx.jpg - Jul 7th 2010, 12:18 AMOpalg
- Jul 7th 2010, 12:27 AMsimplependulum
The solution is

Take modulo we have

otherwise if is odd we have which is impossible .

Thus we have

which leads to

where

which is the multiple of if which is again impossible .

Thus ( not because ) but they can't be both zero as we are looking for positive integers

We now show that is the only solution .

Suppose so we have

If we write down the first six powers of modulo , we obtain :

we have exactly and exactly

We conclude that for some non-negative integers

Back to this equation :

write we have

the multiple of which is also impossible .

Therefore ,

is the only hope , luckily , we obtain and , is the only solution . - Jul 7th 2010, 06:18 AMchiph588@