Self referential quartic

Number Theory Level 5

I have written a quartic polynomial of the form $x^4 - Ax^3 + Bx^2 - Cx + C$ , where $A, B, C$ are positive integers. The roots of the quartic are positive integers, not all necessarily distinct.

Find the maximum possible value of $C$ .

Inspiration.

The answer is 1764.

1 solution

Steven Yuan
May 23, 2017

Let $a, b, c, d$ be the roots of the quartic. From Vieta's, $D = abcd$ and $C = abc + abd + acd + bcd$ . Since $D = C$ , we can write

$abc + abd + acd + bcd = abcd.$

We can divide both sides by $abcd$ (since none of the roots are zero) to get

$\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} + \dfrac{1}{d} = 1.$

Solving this equation requires a lot of work that would greatly expand the length of this solution, so we'll skip it for now. The solutions in positive integers to the equation are

$\begin{aligned} (a, b, c, d) &= (2, 3, 12, 12), (2, 3, 10, 15), (2, 3, 9, 18), \\ &\,\,\,\,(2, 3, 8, 24), (2, 3, 7, 42), (2, 4, 8, 8) \\ &\,\,\,\,(2, 4, 6, 12), (2, 4, 5, 20), (2, 5, 5, 10) \\ &\,\,\,\,(2, 6, 6, 6), (3, 3, 4, 12), (3, 3, 6, 6) \\ &\,\,\,\,(3, 4, 4, 6), (4, 4, 4, 4). \end{aligned}$

Thus, the maximum possible value of $C$ is $\boxed{1764}$ , attained when $(a, b, c, d) = (2, 3, 7, 42)$ .

I think it's much better to ask the question directly. It takes quite a while to phrase what you're saying, at which point in time I lose interest.

Note that the question in brackets isn't identical to the one that you asked. What you want is, "When the coefficient of the constant term is negative of the coefficient of the linear term, what is the maximum possible value of this term?"

Calvin Lin Staff - 4 years ago

I'm not sure what's meant by "ask the question directly," since I was just phrasing it based off of the inspiration. But, let me see if I can formulate it better.

Your second point is valid. Perhaps I do need to edit the question to make it clearer.

Steven Yuan - 4 years ago

Simply say.

Suppose that $a, b, c, d$ are 4 (not necessarily distinct) positive integers such that
$abc+bcd+cda+dab = abcd$
What is the maximum possible value of $abcd$ ?

In general, the more hoops you place for people to jump through, the harder it is for them to follow (and for you to ensure that those are indeed the right hoops).

Calvin Lin Staff - 4 years ago

@Calvin Lin – OK, that makes more sense now. I'm going to see if I can reformulate the problem to be clearer what it's asking for without removing the premise of the quartic polynomial.

Steven Yuan - 4 years ago

Self referential quartic

The answer is 1764.

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