Is cubic a prime?

Number Theory Level 5

$\Large x^3-44x^2+556x-1805$

Let $x_1,x_2,x_3,\dots x_n$ be the positive integers that make the above expression a (positive) prime number. Then, find the value of $\displaystyle \sum_{i=1}^n x_i$ .

Also try Is Quartic a square?

The answer is 30.

1 solution

Nihar Mahajan
May 5, 2015

$x^3-44x^2+556x-1805 \\ \Rightarrow (x-5)(x^2-39x+361)$

For the expression to be prime , one of its factors must be $1$ .

$\rightarrow x-5=1 \Rightarrow \boxed{x = 6} \\ \rightarrow x^2-39x+361=1 \Rightarrow x^2-39x+360=0 \\ \Rightarrow (x-15)(x-24)=0 \Rightarrow \boxed{x=(15,24)}$

When $x=15$ , the polynomial becomes $10$ and is not a prime.

So, required sum $= 6+24= \large\boxed{30}$

Moderator note:

This solution is incomplete.

You still have to check if one of the expressions is equal to -1, and the other is a negative prime.

I got 33... I believe 15 isn't an answer since $f(15)=10$ isn't prime.

Also, to make the solution complete, you should think about one of the factors being $-1$ .

Otto Bretscher - 6 years, 1 month ago

Oh yeah! its indeed 10.Sir please help me where I have gone wrong?

Nihar Mahajan - 6 years, 1 month ago

You say "For the expression to be prime, one of the factors must be 1" (or -1). True, that is necessary, but not sufficient. If one of the factors is $\pm1$ , the expression may or may not be prime, depending on the other factor.You need to check each case.

Otto Bretscher - 6 years, 1 month ago

When you put x = 15 then the quadratic becomes 1 but the linear factor x-5 becomes 10 which makes it a non prime.

You can edit the question by asking $\displaystyle \sum_{i=1}^{n}{x_i} + 18$

Rajdeep Dhingra - 6 years, 1 month ago

@Rajdeep Dhingra – Yes, I have edited the question.

Nihar Mahajan - 6 years, 1 month ago

Well , I appropriately edited the question and my solution.Thanks for your report.

Nihar Mahajan - 6 years, 1 month ago

Also I wish you should have been awarded the 100 points that you lost due to my mistake.

Nihar Mahajan - 6 years, 1 month ago

I'm not in this for the points, but for the fun ;) No worries.

Otto Bretscher - 6 years, 1 month ago

@Otto Bretscher – Wow Sir! I am really very impressed by your spirit! Great, sir!

Mehul Arora - 6 years, 1 month ago

@Mehul Arora – Cheers! xD

Nihar Mahajan - 6 years, 1 month ago

@Otto Bretscher – Sir , can we post a note where we can share this experience , because not me but many people like may tend to not to consider the cases of $\pm 1$ and may go wrong in future.

Nihar Mahajan - 6 years, 1 month ago

@Nihar Mahajan – I'm a lil busy with work right now, but I trust that you can do it. Just write "For the expression to be prime, it is necessary that one of the factors be 1 or -1"... then examine all the cases as you did, but add x=4 (you don't get integers when the other factor is -1).

At the end say: "Now let's see which of these cases produce a prime", and list them all, $f(6)=163, f(15)=10, f(24)= 19, f(4) =-221$ and pick the ones that give you primes, 6 and 24. Hope that helps...

Otto Bretscher - 6 years, 1 month ago

@Otto Bretscher – Thanks for your help!

Nihar Mahajan - 6 years, 1 month ago

Thanks. Those who answered 33 have been marked correct.

I have rephrased the question, and the correct answer is now 30.

In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the “dot dot dot” menu in the lower right corner. This will notify the problem creator who can fix the issues.

Calvin Lin Staff - 6 years, 1 month ago

Why did you tell to add 18 to the final answer?

Anik Mandal - 6 years, 1 month ago

Actually , I had got a wrong answer first.

Nihar Mahajan - 6 years, 1 month ago

Counting the wrong answer you would have got 63?

Anik Mandal - 6 years, 1 month ago

Is cubic a prime?

Also try Is Quartic a square?

The answer is 30.

1 solution

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