If you include 1, there are three.
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They are from 11 squared to 22 squared making a total of 12 square numbers between 100 and 500
To determine the number of prime numbers between 1 and 8888888888888888888888888888888888888888888888, we can use the Prime Number Theorem. This theorem states that the density of prime numbers around a large number n is approximately 1/ln(n). Therefore, the number of prime numbers between 1 and 8888888888888888888888888888888888888888888888 can be estimated by dividing ln(8888888888888888888888888888888888888888888888) by ln(2), which gives approximately 1.33 x 10^27 prime numbers.
Twice. a*a is a squared.
Probably infinite. The smallest is known as Ramanujan's number, 1729 using cubes of 1 and 12, and of 9 and10. Read more about the genius Ramanujan in Wikipaedia.
There are no two "the" numbers. There are infinitely many numbers between 511 and 515. In fact, there are as many numbers between 511 and 515 as there are between 1 and 1000000000000000.