2005 USAMO Problems/Problem 2
(Răzvan Gelca) Prove that the system has no solutions in integers , , and .
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
\[(x^3+y+1)^2 - 1 + z^9 &\equiv -6,\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
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.
Note that the given can be rewritten as
We can also see that
Now we notice
for some pair of non-negative integers . We also note that
when . Furthermore, notice that
for a pair of positive integers means that
which cannot be true. We now know that
which is a contradiction. Now suppose that
We now apply the lifting the exponent lemma to examine the power of 3 that divides each side of the equation to obtain
We can see that 7 must divide m and m-1 which cannot be true as they are relatively prime leading us to conclude that there are no solutions to the given system of diophantine equations.
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