The smallest positive integer ( n ) such that ( n = a^k + b^k ) where ( a ) and ( b ) are positive integers and ( k ) is a positive integer is 2. This occurs when ( a = b = 1 ) and ( k = 1 ), resulting in ( n = 1^1 + 1^1 = 2 ). There are no smaller positive integers that satisfy this equation.
-
Understand the Problem:
We need to find the smallest positive integer ( n ) that can be expressed as the sum of two positive integers raised to the same positive integer power.
-
Identify Constraints:
- ( n ), ( a ), and ( b ) must be positive integers.
- ( k ) is also a positive integer.
-
Start with the Smallest Possible Values:
- Begin with the smallest positive integers for ( a ) and ( b ), which is 1.
- Let ( a = 1 ) and ( b = 1 ).
-
Calculate ( n ):
- ( n = 1^k + 1^k )
- For ( k = 1 ), ( n = 1 + 1 = 2 )
-
Check for Smaller ( n ):
Since 1 is the smallest positive integer, and ( a ) and ( b ) being 1 yields ( n=2 ), there are no smaller ( n ) values that satisfy the equation.
-
Conclusion:
The smallest positive integer ( n ) is 2, achieved when ( a = b = 1 ) and ( k = 1 ).
Final Answer:
The smallest positive integer ( n ) is 2.
