Is n × n - 1 Always a Perfect Square for Any Positive Integer n?

Many believe that when you multiply a positive integer n by itself and subtract 1, the result will always be a perfect square. However, this is not true. We will explore why and provide examples to support this claim.

Understanding Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. For example, 1, 4, 9, 16, etc., are perfect squares because they can be written as 12, 22, 32, 42, and so on.

Is n × (n-1) Always a Perfect Square?

No, n × (n-1) is not always a perfect square for any positive integer n. Let's explore why this is the case through mathematical analysis and specific examples.

Mathematical Analysis

Consider the expression ( n^2 - 1 ). This expression can be factored as ( (n-1)(n 1) ). For ( n^2 - 1 ) to be a perfect square, both ( n-1 ) and ( n 1 ) must be equal because the only way to get a perfect square as a product of two consecutive integers is if they are the same.

Let's examine this in more detail:

For ( n 1 ), ( n^2 - 1 1^2 - 1 0 ). This is indeed a perfect square (02 0). For ( n 2 ), ( n^2 - 1 2^2 - 1 3 ). This is not a perfect square. For ( n 3 ), ( n^2 - 1 3^2 - 1 8 ). This is not a perfect square. For ( n 4 ), ( n^2 - 1 4^2 - 1 15 ). This is not a perfect square.

As you can see, most of the time, ( n^2 - 1 ) is not a perfect square. This is because the difference between ( n^2 ) and ( (n-1)^2 ) is 2n - 1, which is not a perfect square for ( n geq 1 ).

Disproof Through Examples

To further illustrate, let's consider a few examples:

If ( n 1 ), then ( n^2 - 1 1^2 - 1 0 ). This is a perfect square. If ( n 2 ), then ( n^2 - 1 2^2 - 1 3 ). This is not a perfect square. If ( n 3 ), then ( n^2 - 1 3^2 - 1 8 ). This is not a perfect square. If ( n 4 ), then ( n^2 - 1 4^2 - 1 15 ). This is not a perfect square.

From these examples, it is clear that for most positive integers ( n ), ( n^2 - 1 ) is not a perfect square.

Conclusion

In conclusion, ( n times (n-1) ) is not always a perfect square for any positive integer ( n ). While ( n 1 ) is a special case where the result is a perfect square, for all other positive integers, ( n^2 - 1 ) will generally not be a perfect square.

This and similar questions in number theory highlight the importance of understanding the properties of integers and perfect squares. If you have any further questions or need additional examples, feel free to ask!