Difference between revisions of "1977 Canadian MO Problems/Problem 3"
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When <math>\displaystyle a=7^4,</math> <math>\displaystyle b^2+b+1=7^3.</math> Solving this quadratic, <math>\displaystyle b = 18 </math>. | When <math>\displaystyle a=7^4,</math> <math>\displaystyle b^2+b+1=7^3.</math> Solving this quadratic, <math>\displaystyle b = 18 </math>. | ||
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+ | {{alternate solutions}} | ||
== See Also == | == See Also == | ||
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+ | * [[1977 Canadian MO Problems]] | ||
+ | * [[1977 Canadian MO]] | ||
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+ | [[Category:Olympiad Number Theory Problems]] |
Revision as of 22:20, 25 July 2006
Problem
is an integer whose representation in base
is
Find the smallest positive integer
for which
is the fourth power of an integer.
Solution
Rewriting in base
for some integer
Because
and
is prime,
Since we want to minimize
we check to see if
works.
When
Solving this quadratic,
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.