Difference between revisions of "2005 USAMO Problems/Problem 2"
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<math>x^3+1 = 3^a7^b</math> | <math>x^3+1 = 3^a7^b</math> | ||
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for some pair of non-negative integers <math>(a,b)</math>. We also note that | for some pair of non-negative integers <math>(a,b)</math>. We also note that | ||
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for a pair of positive integers <math>(j,k)</math> means that | for a pair of positive integers <math>(j,k)</math> means that | ||
− | <math>x^2-x+1 \ | + | <math>x^2-x+1 \le 3</math> |
which cannot be true. We now know that | which cannot be true. We now know that |
Revision as of 19:01, 23 March 2012
Contents
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
\[(x^3+y+1)^2 - 1 + z^9 &\equiv -6,\] (Error compiling LaTeX. Unknown error_msg)
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.
Solution 2
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
when . Furthermore, notice that
for a pair of positive integers means that
which cannot be true. We now know that
.
Suppose 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.
See also
- <url>Forum/viewtopic.php?p=213009#213009 Discussion on AoPS/MathLinks</url>
2005 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |