Day 23: A Mysterious Polynomial

Algebra Level 4

Let $P(x)$ be a polynomial with integer coefficients.

For a suitable integer $k$ , what is the minimum possible positive difference between $P(k)$ and $P(k+23)$ ?

This problem is part of the Advent Calendar 2015 .

The answer is 23.

2 solutions

Michael Ng
Dec 22, 2015

There is a well known theorem (which is not too difficult to prove) that states that for a polynomial with integer coefficients $P$ and two integers $a$ and $b$ , $a-b \mid P(a)-P(b)$

Applying this we get $k+23-k \mid P(x+23)-P(x) \\ 23 \mid P(x+23)-P(x)$

And so the smallest possible positive difference is $23$ . But we must validate that such a polynomial exists, and in fact $P(x)=x$ works just fine. So the answer is indeed $\boxed{23}$ as required.

Nice. What is the name of the theorem?

First Last - 5 years, 5 months ago

I have the link .

Sahil Nare - 5 years, 5 months ago

For being divisible by 23, why can't the difference be 0?

Manish Mayank - 5 years, 5 months ago

That is because 0 is not a positive number, and the question would like the smallest positive difference. Hope this helps!

Michael Ng - 5 years, 5 months ago

Oh yes! I missed that.

Manish Mayank - 5 years, 5 months ago

Jonathan Alvaro
Dec 28, 2015

This question is a bit off in my opinion. What if the polynomial is of degree zero, that is:

f'(x) = C

With C being an arbitrary constant real number. Wouldn't the difference be 0?

I think that the question is correct. 0 is not a positive number, and the question would like the smallest positive difference. Hope this helps!

Michael Ng - 5 years, 5 months ago

Day 23: A Mysterious Polynomial

This problem is part of the Advent Calendar 2015 .

The answer is 23.

2 solutions

0 pending reports