2005 USAMO Problems/Problem 2
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Problem
(Răzvan Gelca) Prove that the system
has no solutions in integers 
, 
, and 
.
Solution
It suffices to show that there are no solutions to this system in the integers mod 19. We note that 
, so 
. For reference, we construct a table of powers of five:
Evidently, then the order of 5 is 9. Hence 5 is the square of a multiplicative generator of the nonzero integers mod 19, so this table shows all nonzero squares mod 19, as well.
It follows that 
, and 
. Thus we rewrite our system thus:
Adding these, we have
or
By Fermat's Little Theorem, the only possible values of 
are 
and 0, so the only possible values of 
are 
, and 
. But none of these are squares mod 19, a contradiction. Therefore the system has no solutions in the integers mod 19. Therefore the solution has no equation in the integers. 
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See also
| 2005 USAMO (Problems • Resources: AoPS | ML) | ||
| Preceded by Problem 1 | 1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
