How Many Numbers Less Than 100 Are Both Perfect Squares and Cubes?

Have you ever wondered how many numbers, smaller than 100, can simultaneously be both perfect squares and perfect cubes? To answer this curiosity, we will explore the mathematical properties of these numbers and how to find them.

Mathematical Exploration

Non-trivial numbers that are both perfect squares and perfect cubes are relatively rare. In fact, they are perfect sixth powers. A number n is a perfect sixth power if it can be expressed as n k^6 where k is an integer. This is because any number that is both a perfect square and a perfect cube can be written in the form of (k^2)^3 (k^3)^2, which implies n k^6.

Examples of Perfect Sixth Powers

Let's walk through a few examples to see how we can find these numbers.

Starting with ( n 1 )

1^6 1. Since 1^6 equals 1^2^3 and 1^3^2, the number 1 is both a perfect square and a perfect cube.

Next, ( n 2 )

2^6 64. This can be written as 4^3 (2^2^3) and 8^2 (2^3^2). Therefore, 64 is also a perfect sixth power.

For ( n 3 )

3^6 729, which is greater than 100. Therefore, there are no more sixth powers less than 100.

Conclusion and Other Mathematical Aspects

The only positive integers less than 100 that are both a perfect square and a perfect cube are 1 and 64. This makes it quite interesting to consider the prime factorization of such numbers and how they contribute to the factorization of larger numbers, such as 100! (100 factorial).

Factorial and Prime Exponent Pairs

Let's explore how to calculate the exponent of a prime factor in the prime factorization of 100!. To get this number in the prime factorization of 100!, we must round each exponent up to the nearest multiple of 6. This technique can be illustrated with a simple expression:

Expression for Prime Exponents in 100!:

displaystyle sum_{k0} left lfloor frac{100}{p^k} right rfloor

Computing this for 100! results in the following list of prime exponent pairs:

2: 297 3: 48 5: 24 7: 16 11: 9 13: 7 17: 5 19: 5 23: 4 29: 3 31: 3 37: 3 41: 3 43: 3 47: 3 53: 2 59: 2 61: 2 67: 2 71: 2 73: 2 79: 2 83: 2 89: 2 97: 1

After rounding up to the nearest multiple of 6, the prime exponent pairs become:

2: 302 3: 48 5: 24 7: 18 11: 12 13: 12 17: 6 19: 6 23: 6 29: 6 31: 6 37: 6 41: 6 43: 6 47: 6 53: 6 59: 6 61: 6 67: 6 71: 6 73: 6 79: 6 83: 6 89: 6 97: 6

The product of these exponents in the prime factorization of 100! is a very large number. To illustrate, using a Python session, the product of these exponents is:

587885351251269764839165382023399596826310851559219585495497712894588963970036422963108430954704873358628054742210002794248599993366345811476974463743368400341594779519730794407046846775179026336636764145359174509953483982329064396263568856020111116737494601731299635446149021696000000000000000000000000

This number has 303 digits, illustrating just how large the prime factorization can be for such a factorial.