MITPrimes 2018 Question 6

Let PP be a polynomial with integer coefficients and at least 33 simple roots. Is it true that P(n)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, 8 months ago

No vote yet
1 vote

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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 mm such that if pmp|m, then p2mp^2|m is called a "powerful" number. The first few powerful numbers are 1,4,8,9,16,25,27,32,36,1,4,8,9,16,25,27,32, 36, \dots

Powerful numbers are always in the form of a2b3a^2b^3 for a,b1a,b \geq 1.

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

Vishruth Bharath - 1 year, 8 months ago

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

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

Vishruth Bharath - 1 year, 8 months ago

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

Aaghaz Mahajan - 1 year, 8 months ago

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@Aaghaz Mahajan I found it on WolfRam

Vishruth Bharath - 1 year, 8 months ago

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

Aaghaz Mahajan - 1 year, 8 months ago

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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?

Vishruth Bharath - 1 year, 8 months ago

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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.......

Aaghaz Mahajan - 1 year, 8 months ago

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