Difference between revisions of "2000 JBMO Problems/Problem 2"
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+ | ==Problem 2== | ||
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+ | Find all positive integers <math>n\geq 1</math> such that <math>n^2+3^n</math> is the square of an integer. | ||
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== Solution == | == Solution == | ||
Revision as of 23:25, 3 December 2018
Problem 2
Find all positive integers such that is the square of an integer.
Solution
After rearranging we get:
Let
we get: or,
Now, it is clear from above that divides . so,
If
so
But
If then increases exponentially compared to so cannot be .
Thus .
Substituting value of above we get:
or this results in only or
Thus or .