Three numbers.
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Perfect squares have an odd number of factors. There are 31 perfect squares less than 1000.
There are two prime numbers with squares between 100 and 300. These prime numbers are 11 and 13. (112 = 121 and 132 = 169.)
Nine of them.
Infinitely many. there are infinitely many numbers between any two numbers.
Take the square root of 1617 and round it up. Take the square root of 5929 and round it down. That gives you the lower and upper range of the numbers which you must square to get the desired perfect squares.