Proving Even Integer Coefficients in Quadratic Polynomials - Homework Question

[lolo94]

Homework Statement

Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n.

Homework Equations

The Attempt at a Solution

I tried different approaches such as analyzing the constants, f(n)-f(0)-f(1).
How do you approach these problems in general?

 

[SammyS]

Homework Statement

Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n.

Homework Equations

The Attempt at a Solution

I tried different approaches such as analyzing the constants, f(n)-f(0)-f(1).
How do you approach these problems in general?

What can you tell about c from knowledge of f(0) ?

 

[lolo94]

What can you tell about c from knowledge of f(0) ?

c=even integer

 

[member 587159]

We can construct a unique parabola using 3 points. Consider the function f$$x → ax^2 + bx + c$$

We know:
f(0) = a₁
f(1) = a₂

a₁ and a₂ are even integers. You can use f(n₁) for the third point. Then you have a parabola through these 3 points. Try this.