Some time later

Number Theory Level pending

Consider the function f : N → N f:\mathbb{N} \rightarrow \mathbb{N} , where N \mathbb{N} is the set of natural numbers. Suppose f f satisfies the following properties:

  • f ( 1 ) = 1 f(1)=1
  • f ( a + b + a b ) = a + b + f ( a b ) f(a+b+ab)=a+b+f(ab) , for a , b ∈ N a,b \in \mathbb{N}

Find the value of f ( 2015 ) f(2015) .


The answer is 2015.

Pablo Padilla by Pablo Padilla , 24, Mexico

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1 solution

Stephen Brown
Dec 28, 2017

Note that f ( x ) = x f(x)=x satisfies the constraints, so f ( 2015 ) = 2015 f(2015)=\boxed{2015}

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For this solution to be valid, you should be able to prove that no other function satisfies the given properties. Can you prove that such conditions imply that f ( x ) = x f(x)=x ?

Pablo Padilla - 3 years, 5 months ago

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