# 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
1 year, 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 $$a,b \geq 1$$.

- 1 year, 2 months ago

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

- 1 year, 2 months ago

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

- 1 year, 2 months ago

@Aaghaz Mahajan I found it on WolfRam

- 1 year, 2 months ago

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

- 1 year, 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?

- 1 year, 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.......

- 1 year, 2 months ago