Inspired by Danish Ahmed

Algebra Level 4

Classify all positive integer values of n n such that

( 1 + x + x 2 ) ∣ ( 1 + x ) n − 1 n − x n ? ( 1 + x + x^2) \mid (1+x) ^n - 1 ^n - x^n ?

In the options, k k is a positive integer.

Inspiration

6 k ± 1 6k\pm1 2 k ± 1 2k\pm1 12 k ± 1 12k\pm1 3 k ± 1 3k\pm1

Calvin Lin by Calvin Lin

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1 solution

Kushal Dey
Apr 14, 2020

The factor x^2+x+1 has the complex cube root of unity, w as one it's factors. Thus putting x=w in the polynomial, (x+1)^n-1-x^n, we get,. (-w^2)^n-1-w^n, which vanishes if n is an odd integer and not a multiple of 3. Thus integers of the form 6n+1 satisfy.

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Oh ooops, turns out that the answer is 6 k ± 1 6k \pm 1 , which is the solution set of your conditions. Let me fix this.

Calvin Lin Staff - 1 year, 2 months ago

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