# Polynomial proof

## 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.

## 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?

## Answers and Replies

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SammyS
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## 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.

## 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) ?

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

Math_QED
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2019 Award
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.