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

#Combinatorics

Easy Math Editor

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Comments

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 $a,b \geq 1$ .

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

Vishruth Bharath - 3 years, 4 months ago

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

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

Vishruth Bharath - 3 years, 4 months ago

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

Aaghaz Mahajan - 3 years, 4 months ago

@Aaghaz Mahajan I found it on WolfRam

Vishruth Bharath - 3 years, 4 months ago

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

Aaghaz Mahajan - 3 years, 4 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?

Vishruth Bharath - 3 years, 4 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.......

Aaghaz Mahajan - 3 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$