To determine if the equation [tex]\( x^n + y^n = z^n \)[/tex] holds for positive integers [tex]\( x \)[/tex], [tex]\( y \)[/tex], [tex]\( z \)[/tex], and [tex]\( n \)[/tex], we need to consider different values of [tex]\( n \)[/tex].
1. When [tex]\( n = 2 \)[/tex]:
- This equation becomes [tex]\( x^2 + y^2 = z^2 \)[/tex].
- This is known as the Pythagorean theorem.
- It states that there are sets of positive integers [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] (known as Pythagorean triples) which satisfy this equation. For example, [tex]\( 3^2 + 4^2 = 5^2 \)[/tex], where [tex]\( 3 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 5 \)[/tex] are positive integers.
2. When [tex]\( n > 2 \)[/tex]:
- According to Fermat's Last Theorem, there are no three positive integers [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy the equation [tex]\( x^n + y^n = z^n \)[/tex] for any integer value of [tex]\( n \)[/tex] greater than 2.
- This means that for [tex]\( n > 2 \)[/tex], there are no sets of positive integers [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that make the equation true.
Hence, among all positive integer values of [tex]\( n \)[/tex], the equation [tex]\( x^n + y^n = z^n \)[/tex] is only true for [tex]\( n = 2 \)[/tex].
So, the answer to the question is:
[tex]\[ n = 2 \][/tex]
This demonstrates that the equation [tex]\( x^n + y^n = z^n \)[/tex] holds true for positive integers [tex]\( x \)[/tex], [tex]\( y \)[/tex], [tex]\( z \)[/tex], and [tex]\( n \)[/tex] if and only if [tex]\( n = 2 \)[/tex].
