Understanding n^n as a Perfect Square: Analyzing Integers Between 1 and 50

Understanding n^n as a Perfect Square: Analyzing Integers Between 1 and 50

In mathematics, the concept of a perfect square is intriguing, especially when applied to powers of integers. Specifically, we are interested in the number of integers n between 1 and 50 (inclusive) such that n^n is a perfect square. This problem requires a blend of number theory and careful enumeration.

An Overview of the Problem

The core of our question is to determine how many integers, n, satisfy the condition that n^n is a perfect square. A number n^n is a perfect square if and only if n is either an even integer or a perfect square. We will explore each of these cases in detail.

Case Analysis to Find Perfect Squares

Case 1: n is an Even Integer

First, consider the scenario where n is an even integer. For any even integer n, n^n can be written as (2k)^n, which simplifies to 2^n k^n. For this to be a perfect square, both 2^n and k^n must result in perfect squares, which is true if n is even. Since even integers are of the form 2k, if we list all even integers between 1 and 50, we get:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50

Counting these, we find there are 25 even integers between 1 and 50, inclusive.

Case 2: n is a Perfect Square

Secondly, consider the scenario where n itself is a perfect square. A perfect square between 1 and 50 can be any integer of the form k^2. The perfect squares between 1 and 50 are:

1, 4, 9, 16, 25, 36, 49

There are 7 such perfect squares. However, we must note that the integer 4, which is both an even integer and a perfect square, is counted in both cases. Therefore, we must subtract this overlap to avoid double-counting.

Combining Both Cases

Now, combining both cases, we identify the total number of integers that satisfy the condition. We have 25 even integers and 7 perfect squares, but since 4 is included in both, we need to subtract this overlap:

Total number of such integers 25 (even integers) 7 (perfect squares) - 1 (overlap at 4) 31.

Conclusion

Therefore, the total number of integers n, between 1 and 50, such that n^n is a perfect square, is 31. This count includes all even integers and perfect squares, minus the integer 4 which was counted twice.

Additional Insights

One might also consider if there are any other integers n such that n^n is a perfect square, apart from the 25 even integers and 7 perfect squares. As a matter of fact, for n 1 and n 50, 1^1 1 and 50^50 (which is not a perfect square in the context of this problem since 50 is not a perfect square and is uneven), there is no other integer between 1 and 50 that fits this condition.

Final Answer

Therefore, the total number of integers n between 1 and 50 such that n^n is a perfect square is 31.