Understanding 4-Digit Integers that are Both Perfect Squares and Perfect Cubes

In the realm of number theory, exploring integers that are both perfect squares and perfect cubes is a fascinating topic. A number that is both a perfect square and a perfect cube is known as a perfect sixth power. This article delves into how many 4-digit positive integers are perfect sixth powers, thus meeting the criteria of being both a perfect square and a perfect cube.

Perfect Sixth Powers

A number that is both a perfect square and a perfect cube must be a perfect sixth power. If we denote such a number as ( n^6 ), where ( n ) is a positive integer, we need to determine the range of ( n ) such that ( n^6 ) is a 4-digit number.

Range Determination

To find the range of ( n ) such that ( 1000 leq n^6 leq 9999 ):

Lower Bound:

To find the smallest ( n ) such that ( n^6 geq 1000 ):

( n geq 1000^{1/6} )

Calculating ( 1000^{1/6} ):

( 1000^{1/6} 10^3^{1/6} 10^{3/6} 10^{1/2} sqrt{10} approx 3.162 )

The smallest integer ( n ) satisfying this is ( n 4 ).

Upper Bound:

To find the largest ( n ) such that ( n^6 leq 9999 ):

( n leq 9999^{1/6} )

Calculating ( 9999^{1/6} ):

( 9999^{1/6} approx 10^3^{1/6} 10^{1/2} approx 3.98 )

The largest integer ( n ) satisfying this is ( n 3 ).

Since ( n ) must be an integer, let's check the values:

( n 4 ) gives ( 4^6 4096 ), which is a 4-digit number. ( n 3 ) gives ( 3^6 729 ), which is not a 4-digit number.

Therefore, the only valid integer ( n ) is 4.

Conclusion

The total number of 4-digit positive integers that are both perfect squares and perfect cubes is:

( boxed{1} )

Thus, only one 4-digit number, 4096, is both a perfect square and a perfect cube.