# MITPrimes 2018 Question 6

Let $P$ be a polynomial with integer coefficients and at least $3$ simple roots. Is it true that $P(n)$ is powerful only finitely often?

Can you guys tell me how you would approach this problem?

Source: MITPrimes 2018

Note by Vishruth Bharath
3 years, 2 months ago

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After doing some research, I've found what a "powerful" number is. Basically, if we let a number be represented by $m$ such that if $p|m$, then $p^2|m$ is called a "powerful" number. The first few powerful numbers are $1,4,8,9,16,25,27,32, 36, \dots$

Powerful numbers are always in the form of $a^2b^3$ for $a,b \geq 1$.

- 3 years, 2 months ago

It's about the $\text{abc}$ conjecture for powerful numbers.

- 3 years, 2 months ago

Ok!! That's some new information....Thanks......Dude where did you find all this??

- 3 years, 2 months ago

@Aaghaz Mahajan I found it on WolfRam

- 3 years, 2 months ago

What do you mean by "Powerful" ?? Also, what are "simple roots" ?? Are they integral roots??

- 3 years, 2 months ago

Same, I have doubts about what makes something "powerful." Also, I believe simple roots are not integral roots. @Chew-Seong Cheong what do you think?

- 3 years, 2 months ago

Well, maybe the question paper had some previously stated criteria for defining these terms........I even checked on the net and couldn't find an aswer to this query.......

- 3 years, 2 months ago