There are infinitely many, just like in base 10.
In any base system, the number of perfect squares is the same.
Take the natural (counting) numbers 1, 2, 3, ....
Squaring each of these produces the perfect squares.
As there are an infinite number of natural numbers, there are an infinite number of perfect squares.
The first 10 perfect squares in base 5 are: 15, 45, 145, 315, 1005, 1215, 1445, 2245, 3115, 4005, ...
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So far, no odd perfect numbers have been found to exist. If and only if one does exist, it will be far larger than the number in the question. In fact, it has been proven concretely that if an odd perfect number exists, it is greater than 10^300, and it is conjectured that if such a number exists, it is greater than 10^500.
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No, it isn't. In fact, it is not known whether are odd perfect numbers exist. The first perfect numbers are 6, 28, 496, and 8128.
Yes, it does exist.
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