# Elementary math proof

## Homework Statement

Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd.

## The Attempt at a Solution

It's a little bit intuitive.
Proof
1 and 3 have the same parity. They are both odd
so if(odd)=odd then f(1)=odd and f(3)=odd
or if(odd)=even then f(1)=even and f(3)=even

is that right?

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd.

## The Attempt at a Solution

It's a little bit intuitive.
Proof
1 and 3 have the same parity. They are both odd
so if(odd)=odd then f(1)=odd and f(3)=odd
or if(odd)=even then f(1)=even and f(3)=even

is that right?

No, it is not correct, because you are essentially assuming what you want to prove. In order to be able to assert that f(odd) = odd, for example, you need to show that it is not possible to have f(odd1) = odd but f(odd2) = even, etc.

Hint: if there was such a polynomial ##f(3)-f(1)## would be both even and odd.

Hint: if there was such a polynomial ##f(3)-f(1)## would be both even and odd.
why would they both be even and odd?

fresh_42
Mentor
2021 Award
why would they both be even and odd?
What do you know about ##f(3) - f(1)##?

haruspex