Difference between revisions of "2000 JBMO Problems/Problem 2"
(Created page with "== Solution == After rearranging we get: <math>(k-n)(k+n) = 3^n</math> Let <math>k-n = 3^a, k+n = 3^{n-a}</math> we get: <math>2n = 3^a(3^{n-2a} - 1)</math> or, <math>(2n/(...") |
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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 00:25, 4 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
.