Inspired by Danish Ahmed - Part 2

Algebra Level 4

Classify all positive integer values of $n$ such that

$( 1 + x + x^2) ^2 \mid (1+x)^n - 1^n - x^n$

1 more than a multiple of 18 only 1 more than a multiple of 6 only 1 more than a multiple of 12 only 1 more than a multiple of 3 only

1 solution

Kushal Dey
Apr 14, 2020

Extending my solution to "Previous Part", whose link is given at the end of the problem, we see that x^2+x+1 divides the polynomial. But for (x^2+x+1)^2 to divide the polynomial it means that x=w is a repeated root of the above. Let P(x)=(1+x)^n-1-x^n. Differentiating, P'(x)=n((1+x)^(n-1)-x^(n-1)). Since n was of the form 6k+1, n-1 = 6k. Thus once again the polynomial is vanishes when we put x=w.

As pointed out from the previous part, we need to verify that $6k - 1$ isn't a solution.

Calvin Lin Staff - 1 year, 2 months ago

Inspired by Danish Ahmed - Part 2

1 solution

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