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

**Source:** MITPrimes 2018

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TopNewestAfter 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 \(a,b \geq 1\).

If you want to read more about "powerful" numbers, visit this link: http://mathworld.wolfram.com/PowerfulNumber.html

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Here is another link: https://arxiv.org/abs/1611.01192

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

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

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@Aaghaz Mahajan I found it on WolfRam

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

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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?

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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.......

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